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logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[71,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[71,logseq____"^Hlogseq____",34,536870922]],[logseq____"^15logseq____",[71,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[71,logseq____"^;logseq____",logseq____"~u652fddf3-3561-4498-9970-ba2f4c7d5f1alogseq____",536870922]],[logseq____"^15logseq____",[72,logseq____"^Qlogseq____",logseq____"## [[Binomialkoeffizient]]logseq____",536870922]],[logseq____"^15logseq____",[72,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[72,logseq____"^Flogseq____",75,536870922]],[logseq____"^15logseq____",[72,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[72,logseq____"^Vlogseq____",105,536870922]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",32,536870922]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",36,536870922]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",38,536870922]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[72,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870922]],[logseq____"^15logseq____",[72,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[72,logseq____"^Hlogseq____",36,536870922]],[logseq____"^15logseq____",[72,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[72,logseq____"^;logseq____",logseq____"~u652fddf3-716c-4885-ab81-7e6f8abb299dlogseq____",536870922]],[logseq____"^15logseq____",[73,logseq____"^Qlogseq____",logseq____"$\\\\text{Alte Schreibweise}$\\n\\\\begin{align}\\n\\\\text{cosa }logseq____&\\\\text{plus }logseq____&\\\\text{cubus }logseq____&\\\\text{acq }logseq____&6\\\\\\\\\\nxlogseq____&+logseq____&x^3logseq____&=logseq____&6\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[73,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[73,logseq____"^Flogseq____",65,536870922]],[logseq____"^15logseq____",[73,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[73,logseq____"^Vlogseq____",94,536870922]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[73,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[73,logseq____"^;logseq____",logseq____"~u652fddf3-4f5d-449e-9631-7f16d3b248d0logseq____",536870922]],[logseq____"^15logseq____",[74,logseq____"^Qlogseq____",logseq____"# Was ist Größer?logseq____",536870922]],[logseq____"^15logseq____",[74,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[74,logseq____"^Flogseq____",79,536870922]],[logseq____"^15logseq____",[74,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[74,logseq____"^Vlogseq____",44,536870922]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[74,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[74,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[74,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[74,logseq____"^;logseq____",logseq____"~u652fddf3-c6c8-43c8-a778-73c85f9cd3a5logseq____",536870922]],[logseq____"^15logseq____",[75,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870922]],[logseq____"^15logseq____",[75,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[75,logseq____"^Flogseq____",69,536870922]],[logseq____"^15logseq____",[75,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[75,logseq____"^Vlogseq____",105,536870922]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",32,536870922]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",38,536870922]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",39,536870922]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[75,logseq____"^Hlogseq____",39,536870922]],[logseq____"^15logseq____",[75,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[75,logseq____"^;logseq____",logseq____"~u652fddf3-8f4c-41c1-b829-abde4f84f06clogseq____",536870922]],[logseq____"^15logseq____",[76,logseq____"^Qlogseq____",logseq____"#FIXME $(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5=2+20x+40x^2$ ist Falschlogseq____",536870922]],[logseq____"^15logseq____",[76,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[76,logseq____"^Flogseq____",66,536870922]],[logseq____"^15logseq____",[76,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[76,logseq____"^Vlogseq____",75,536870922]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",32,536870922]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",38,536870922]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",39,536870922]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",46,536870922]],[logseq____"^15logseq____",[76,logseq____"^Hlogseq____",46,536870922]],[logseq____"^15logseq____",[76,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[76,logseq____"^;logseq____",logseq____"~u652fddf3-6e3b-4e0a-8af1-5248205c864flogseq____",536870922]],[logseq____"^15logseq____",[77,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche Gleichungen sind richtig?}$\\n$\\\\text{Wenn } 10^x=y \\\\text{, dann}$\\n\\\\begin{align}\\nxlogseq____&=\\\\lg y logseq____& ,r\\\\\\\\\\nxlogseq____&=\\\\log_y10logseq____&,f\\\\\\\\\\nxlogseq____&=\\\\log_{10}ylogseq____&,r \\\\\\\\\\n\\\\ln 4 - \\\\ln 2 logseq____&= \\\\ln 2 logseq____&, r \\\\\\\\\\n\\\\ln 4 - \\\\ln 2 logseq____&= \\\\ln 8 logseq____&, f \\\\\\\\\\n\\\\ln 4 + \\\\ln 2 logseq____&= \\\\ln 8 logseq____&, r \\\\\\\\\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[77,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[77,logseq____"^Flogseq____",86,536870922]],[logseq____"^15logseq____",[77,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[77,logseq____"^Vlogseq____",86,536870922]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[77,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[77,logseq____"^;logseq____",logseq____"~u652fddf3-946c-44cc-94a3-a0415a2c055flogseq____",536870922]],[logseq____"^15logseq____",[78,logseq____"^Qlogseq____",logseq____"## [[Rationalmachen]] des [[Nenner]]slogseq____",536870922]],[logseq____"^15logseq____",[78,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[78,logseq____"^Flogseq____",70,536870922]],[logseq____"^15logseq____",[78,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[78,logseq____"^Vlogseq____",61,536870922]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",43,536870922]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",49,536870922]],[logseq____"^15logseq____",[78,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870922]],[logseq____"^15logseq____",[78,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[78,logseq____"^Hlogseq____",43,536870922]],[logseq____"^15logseq____",[78,logseq____"^Hlogseq____",49,536870922]],[logseq____"^15logseq____",[78,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[78,logseq____"^;logseq____",logseq____"~u652fddf3-848c-4259-9556-aff6e399fd5blogseq____",536870922]],[logseq____"^15logseq____",[79,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Wurzeln]]logseq____",536870922]],[logseq____"^15logseq____",[79,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[79,logseq____"^Flogseq____",86,536870922]],[logseq____"^15logseq____",[79,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[79,logseq____"^Vlogseq____",44,536870922]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[79,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[79,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[79,logseq____"^Hlogseq____",41,536870922]],[logseq____"^15logseq____",[79,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[79,logseq____"^;logseq____",logseq____"~u652fddf3-9e4c-4d1f-9494-f190df60ff99logseq____",536870922]],[logseq____"^15logseq____",[80,logseq____"^Qlogseq____",logseq____"[[Wolframalpha]]logseq____",536870922]],[logseq____"^15logseq____",[80,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[80,logseq____"^Flogseq____",82,536870922]],[logseq____"^15logseq____",[80,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[80,logseq____"^Vlogseq____",82,536870922]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",48,536870922]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",50,536870922]],[logseq____"^15logseq____",[80,logseq____"^Hlogseq____",50,536870922]],[logseq____"^15logseq____",[80,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[80,logseq____"^;logseq____",logseq____"~u652fddf3-c060-4907-8967-705a406f9a7flogseq____",536870922]],[logseq____"^15logseq____",[81,logseq____"^Qlogseq____",logseq____"[[Satz von Vieta]]\\n\\\\begin{align}\\nx^2+px+qlogseq____&=0 logseq____&logseq____& \\\\text{habe }x_{1,2}\\\\text{ als Lösung}\\\\\\\\\\nx_1+x_2logseq____&=-p logseq____&logseq____& x_1*x_2=q\\\\\\\\\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[81,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[81,logseq____"^Flogseq____",51,536870922]],[logseq____"^15logseq____",[81,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[81,logseq____"^Vlogseq____",63,536870922]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",33,536870922]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",45,536870922]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",47,536870922]],[logseq____"^15logseq____",[81,logseq____"^Hlogseq____",45,536870922]],[logseq____"^15logseq____",[81,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[81,logseq____"^;logseq____",logseq____"~u652fddf3-278f-4223-a697-74eaae1b3241logseq____",536870922]],[logseq____"^15logseq____",[82,logseq____"^Qlogseq____",logseq____"# [[Links]]logseq____",536870922]],[logseq____"^15logseq____",[82,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[82,logseq____"^Flogseq____",44,536870922]],[logseq____"^15logseq____",[82,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[82,logseq____"^Vlogseq____",44,536870922]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",48,536870922]],[logseq____"^15logseq____",[82,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[82,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[82,logseq____"^Hlogseq____",48,536870922]],[logseq____"^15logseq____",[82,logseq____"^17logseq____",false,536870922]],[logseq____"^15logseq____",[82,logseq____"^;logseq____",logseq____"~u652fddf3-c468-406a-a249-7f7a8f099112logseq____",536870922]],[logseq____"^15logseq____",[83,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn ergibtlogseq____",536870922]],[logseq____"^15logseq____",[83,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[83,logseq____"^Flogseq____",57,536870922]],[logseq____"^15logseq____",[83,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[83,logseq____"^Vlogseq____",84,536870922]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",39,536870922]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",46,536870922]],[logseq____"^15logseq____",[83,logseq____"^Hlogseq____",46,536870922]],[logseq____"^15logseq____",[83,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[83,logseq____"^;logseq____",logseq____"~u652fddf3-b5e0-4ea8-b9c7-85cfe8557b90logseq____",536870922]],[logseq____"^15logseq____",[84,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870922]],[logseq____"^15logseq____",[84,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[84,logseq____"^Flogseq____",64,536870922]],[logseq____"^15logseq____",[84,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[84,logseq____"^Vlogseq____",61,536870922]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",39,536870922]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[84,logseq____"^Hlogseq____",39,536870922]],[logseq____"^15logseq____",[84,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[84,logseq____"^;logseq____",logseq____"~u652fddf3-ade0-4bd9-9776-42bb061ec6dclogseq____",536870922]],[logseq____"^15logseq____",[85,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870922]],[logseq____"^15logseq____",[85,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[85,logseq____"^Flogseq____",81,536870922]],[logseq____"^15logseq____",[85,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[85,logseq____"^Vlogseq____",81,536870922]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",33,536870922]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",39,536870922]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",45,536870922]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",47,536870922]],[logseq____"^15logseq____",[85,logseq____"^Hlogseq____",39,536870922]],[logseq____"^15logseq____",[85,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[85,logseq____"^;logseq____",logseq____"~u652fddf3-35e3-45b0-bdb1-e1c0427a7038logseq____",536870922]],[logseq____"^15logseq____",[86,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Logarithmen]]logseq____",536870922]],[logseq____"^15logseq____",[86,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[86,logseq____"^Flogseq____",87,536870922]],[logseq____"^15logseq____",[86,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[86,logseq____"^Vlogseq____",44,536870922]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[86,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[86,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[86,logseq____"^Hlogseq____",35,536870922]],[logseq____"^15logseq____",[86,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[86,logseq____"^;logseq____",logseq____"~u652fddf3-e156-41d3-b542-381a9261a760logseq____",536870922]],[logseq____"^15logseq____",[87,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Potenzen]]logseq____",536870922]],[logseq____"^15logseq____",[87,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[87,logseq____"^Flogseq____",82,536870922]],[logseq____"^15logseq____",[87,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[87,logseq____"^Vlogseq____",44,536870922]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[87,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[87,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[87,logseq____"^Hlogseq____",31,536870922]],[logseq____"^15logseq____",[87,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[87,logseq____"^;logseq____",logseq____"~u652fddf3-be4e-43eb-8d05-3e3305ca7287logseq____",536870922]],[logseq____"^15logseq____",[88,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\sqrt{a+5}-\\\\sqrt{x+3}logseq____&=2\\\\sqrt(x+1) logseq____&|logseq____&()^2\\\\\\\\\\n\\\\left(\\\\sqrt{x+5}-\\\\sqrt{x+3}\\\\right)^2logseq____&=4(x+1)\\\\\\\\\\n(x+5)-2\\\\sqrt{x+5}\\\\sqrt{x+3}+(x+3)logseq____&=4x+4logseq____&|logseq____&-(2x+8)\\\\\\\\\\n-2\\\\sqrt{(x+5)(x+3)}logseq____&=2x-4 logseq____&|logseq____&()^2\\\\\\\\\\n4(x+5)(x+3)logseq____&=4x^2-16x+16\\\\\\\\\\n4x^2+32x+60logseq____&=4x^2-16x+16logseq____&|logseq____&-4x^2+16x-16\\\\\\\\\\n48x+44logseq____&=0\\\\quad x=-\\\\frac{11}{12}\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[88,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[88,logseq____"^Flogseq____",71,536870922]],[logseq____"^15logseq____",[88,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[88,logseq____"^Vlogseq____",71,536870922]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",34,536870922]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[88,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[88,logseq____"^;logseq____",logseq____"~u652fddf3-c98b-4e20-a754-ec283eda39dflogseq____",536870922]],[logseq____"^15logseq____",[89,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche quadratische Gleichung hat die Lösung}$\\n$x_1=3\\\\quad x_2=4$\\n\\\\begin{align}\\nx_1+x_2 = 7 logseq____&= p logseq____&, x_1x_2=12=q\\\\\\\\\\nx^2+px+qlogseq____&=0\\\\\\\\\\nx^2-7x+12logseq____&=0\\\\\\\\\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[89,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[89,logseq____"^Flogseq____",85,536870922]],[logseq____"^15logseq____",[89,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[89,logseq____"^Vlogseq____",85,536870922]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",33,536870922]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",39,536870922]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",45,536870922]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",47,536870922]],[logseq____"^15logseq____",[89,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[89,logseq____"^;logseq____",logseq____"~u652fddf3-14ce-4313-843f-c14eb8d30c83logseq____",536870922]],[logseq____"^15logseq____",[90,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])\\n\\\\begin{align}\\nx^4-34x^2+225logseq____&=0\\\\\\\\\\n\\\\text{substituiert }ylogseq____&=x^2\\\\\\\\\\ny^2-34y+225logseq____&=0\\\\\\\\\\ny_{1,2}logseq____&=17\\\\pm\\\\sqrt{(-17)^2-225}\\\\\\\\\\nlogseq____&=17\\\\pm\\\\sqrt{289-225}\\nlogseq____&=17\\\\pm\\\\sqrt{64}=17\\\\pm8\\\\\\\\\\nlogseq____&=\\\\left.\\\\begin{cases}\\n25\\\\\\\\\\n9\\n\\\\end{cases}\\\\right|\\\\begin{array}{l}\\nx^2=25logseq____&\\\\rightarrow logseq____&x_1=5;logseq____&x_2=-5\\\\\\\\\\nx^2=9logseq____&\\\\rightarrow logseq____&x_1=3;logseq____&x_2=-3\\\\\\\\\\n\\\\end{array}\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[90,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[90,logseq____"^Flogseq____",81,536870922]],[logseq____"^15logseq____",[90,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[90,logseq____"^Vlogseq____",63,536870922]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",33,536870922]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",39,536870922]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",47,536870922]],[logseq____"^15logseq____",[90,logseq____"^Hlogseq____",39,536870922]],[logseq____"^15logseq____",[90,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[90,logseq____"^;logseq____",logseq____"~u652fddf3-ee06-40d3-b1cd-e2133a3c5732logseq____",536870922]],[logseq____"^15logseq____",[91,logseq____"^Qlogseq____",logseq____"cosa -logseq____> beliebige Variable\\ncubus -logseq____> hoch 3logseq____",536870922]],[logseq____"^15logseq____",[91,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[91,logseq____"^Flogseq____",73,536870922]],[logseq____"^15logseq____",[91,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[91,logseq____"^Vlogseq____",73,536870922]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[91,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[91,logseq____"^;logseq____",logseq____"~u652fddf3-1c06-456d-a4a7-bf9f3fb370b1logseq____",536870922]],[logseq____"^15logseq____",[92,logseq____"^Qlogseq____",logseq____"$\\\\text{Wenn }alogseq____>0\\\\text{, dann}$\\n\\\\begin{align}\\n\\\\frac b{\\\\sqrt a} = \\\\frac b{\\\\sqrt a}*\\\\frac{\\\\sqrt a}{\\\\sqrt a} = \\\\frac {b\\\\sqrt a}a\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[92,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[92,logseq____"^Flogseq____",78,536870922]],[logseq____"^15logseq____",[92,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[92,logseq____"^Vlogseq____",78,536870922]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",43,536870922]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",49,536870922]],[logseq____"^15logseq____",[92,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[92,logseq____"^;logseq____",logseq____"~u652fddf3-5d16-47b0-8c13-0d82af410267logseq____",536870922]],[logseq____"^15logseq____",[93,logseq____"^Qlogseq____",logseq____"# [[Quadratische Gleichungen]]logseq____",536870922]],[logseq____"^15logseq____",[93,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[93,logseq____"^Flogseq____",61,536870922]],[logseq____"^15logseq____",[93,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[93,logseq____"^Vlogseq____",99,536870922]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",33,536870922]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[93,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[93,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[93,logseq____"^Hlogseq____",33,536870922]],[logseq____"^15logseq____",[93,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[93,logseq____"^;logseq____",logseq____"~u652fddf3-40ce-4373-8441-20f2c2951779logseq____",536870922]],[logseq____"^15logseq____",[94,logseq____"^Qlogseq____",logseq____"# [[Potenzen]]logseq____",536870922]],[logseq____"^15logseq____",[94,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[94,logseq____"^Flogseq____",99,536870922]],[logseq____"^15logseq____",[94,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[94,logseq____"^Vlogseq____",99,536870922]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[94,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[94,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[94,logseq____"^Hlogseq____",31,536870922]],[logseq____"^15logseq____",[94,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[94,logseq____"^;logseq____",logseq____"~u652fddf3-7ca2-4446-9378-ddd091d9fb45logseq____",536870922]],[logseq____"^15logseq____",[95,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\na^nlogseq____&=\\\\underbrace{a*a*\\\\cdots*n}_n logseq____& a,blogseq____>0\\\\\\\\\\na^2b^2logseq____&=(ab)^n\\\\\\\\\\n\\\\frac{a^n}{b^n}logseq____&=\\\\left(\\\\frac ab\\\\right)^n\\\\\\\\\\na^na^mlogseq____&=a^{n+m}\\\\\\\\\\n\\\\frac1{a^n}logseq____&=a^{-n}\\\\\\\\\\n(a^n)^mlogseq____&=a^{nm}\\\\\\\\\\na^0logseq____&=1\\\\\\\\\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[95,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[95,logseq____"^Flogseq____",94,536870922]],[logseq____"^15logseq____",[95,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[95,logseq____"^Vlogseq____",94,536870922]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[95,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[95,logseq____"^;logseq____",logseq____"~u652fddf3-5ff7-47fa-90c7-51997a3b9974logseq____",536870922]],[logseq____"^15logseq____",[96,logseq____"^Qlogseq____",logseq____"Einsetzen, um zu überprüfen, ob man eine Fallunterscheidung brauchtlogseq____",536870922]],[logseq____"^15logseq____",[96,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[96,logseq____"^Flogseq____",88,536870922]],[logseq____"^15logseq____",[96,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[96,logseq____"^Vlogseq____",71,536870922]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",34,536870922]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[96,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[96,logseq____"^;logseq____",logseq____"~u652fddf3-6c7e-4ca5-879f-4eeb9c7c79e0logseq____",536870922]],[logseq____"^15logseq____",[97,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche Gleichungen sind richtig?}$\\n\\\\begin{align}\\n\\\\tan x logseq____&= \\\\frac{\\\\sin x}{\\\\cos x} logseq____&, \\\\cos x \\\\ne 0\\\\\\\\\\n\\\\sin^{-2}x-\\\\cos^2xlogseq____&=1logseq____&,f\\\\\\\\\\n\\\\sin x = \\\\cos x logseq____&= 1 logseq____&, f\\\\\\\\\\n\\\\sin^{-2}x+cos^2ylogseq____&=1logseq____&,r\\\\\\\\\\n1+tan^2xlogseq____&=\\\\frac1{\\\\cos^2x}logseq____&,\\\\cos x\\\\ne0\\\\\\\\\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[97,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[97,logseq____"^Flogseq____",68,536870922]],[logseq____"^15logseq____",[97,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[97,logseq____"^Vlogseq____",54,536870922]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",40,536870922]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[97,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[97,logseq____"^;logseq____",logseq____"~u652fddf3-d72a-4d90-8a35-ee170f12f028logseq____",536870922]],[logseq____"^15logseq____",[98,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiel für die 3. Binomische Formel}$\\n\\\\begin{align}\\n\\\\frac{\\\\frac1{a-b}+\\\\frac1{a+b}}{\\\\frac1{a-b}-\\\\frac1{a+b}}\\nlogseq____&=\\\\frac{\\\\frac{a+b+a-b}{(a+b)*(a-b)}}{\\\\frac{a+b-(a-b)}{(a+b)*(a-b)}}\\nlogseq____&=\\\\frac{a+b+a-b}{a+b-a+b}\\nlogseq____&=\\\\frac{2a}{2b}\\nlogseq____&=\\\\frac ab\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[98,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[98,logseq____"^Flogseq____",67,536870922]],[logseq____"^15logseq____",[98,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[98,logseq____"^Vlogseq____",58,536870922]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",38,536870922]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[98,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[98,logseq____"^;logseq____",logseq____"~u652fddf3-272b-4342-994f-28439befc592logseq____",536870922]],[logseq____"^15logseq____",[99,logseq____"^Qlogseq____",logseq____"# [[Potenzen]] und [[Wurzeln]]logseq____",536870922]],[logseq____"^15logseq____",[99,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[99,logseq____"^Flogseq____",58,536870922]],[logseq____"^15logseq____",[99,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[99,logseq____"^Vlogseq____",44,536870922]],[logseq____"^15logseq____",[99,logseq____"^Ulogseq____",31,536870922]],[logseq____"^15logseq____",[99,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[99,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[99,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[99,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[99,logseq____"^Hlogseq____",31,536870922]],[logseq____"^15logseq____",[99,logseq____"^Hlogseq____",41,536870922]],[logseq____"^15logseq____",[99,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[99,logseq____"^;logseq____",logseq____"~u652fddf3-0b03-46fd-b121-ded1951db1e2logseq____",536870922]],[logseq____"^15logseq____",[100,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche Gleichungen sind richtig?}$\\n$\\\\text{Wenn }alogseq____>0\\\\text{, dann}$\\n\\\\begin{align}\\n\\\\frac{a^2}{\\\\sqrt{a}}logseq____&=a\\\\sqrt{a}logseq____&,r\\\\\\\\\\n\\\\frac{a^2}{\\\\sqrt{a}}logseq____&=a^2-\\\\sqrt{a}logseq____&,f\\\\\\\\\\n\\\\frac{a^2}{\\\\sqrt{a}}logseq____&=\\\\sqrt{a^3}logseq____&,r\\\\\\\\\\n\\\\frac{a^2}{\\\\sqrt{a}}logseq____&=a^{\\\\frac32}logseq____&,r\\\\\\\\\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[100,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[100,logseq____"^Flogseq____",79,536870922]],[logseq____"^15logseq____",[100,logseq____"^Xlogseq____",44,536870922]],[logseq____"^15logseq____",[100,logseq____"^Vlogseq____",79,536870922]],[logseq____"^15logseq____",[100,logseq____"^Ulogseq____",41,536870922]],[logseq____"^15logseq____",[100,logseq____"^Ulogseq____",44,536870922]],[logseq____"^15logseq____",[100,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[100,logseq____"^;logseq____",logseq____"~u652fddf3-b545-4f5d-8054-36673821152blogseq____",536870922]],[logseq____"^15logseq____",[101,logseq____"^Qlogseq____",logseq____"Das [[Pascalsche Dreieck]] lässt sich auch mit den entsprechenden Binomialkoeffizienten 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x=\\\\sqrt[n]a$ #FIXME ich glaube hier fehlt logseq____>0 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Lösung mit [[Fallunterscheidung]]logseq____",536870923]],[logseq____"^15logseq____",[150,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[150,logseq____"^Flogseq____",164,536870923]],[logseq____"^15logseq____",[150,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[150,logseq____"^Vlogseq____",164,536870923]],[logseq____"^15logseq____",[150,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[150,logseq____"^Ulogseq____",112,536870923]],[logseq____"^15logseq____",[150,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[150,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[150,logseq____"^Ulogseq____",119,536870923]],[logseq____"^15logseq____",[150,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[150,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870923]],[logseq____"^15logseq____",[150,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[150,logseq____"^Hlogseq____",112,536870923]],[logseq____"^15logseq____",[150,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[150,logseq____"^;logseq____",logseq____"~u652fddf3-8f28-4f4a-8e3c-8902fd0e4983logseq____",536870923]],[logseq____"^15logseq____",[151,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 1} \\\\quad xlogseq____<-5$\\n\\\\begin{align}\\nx+5logseq____&logseq____<0\\\\\\\\\\n|x+5|logseq____&=-(x+5)=-x-5\\\\\\\\\\nx-3logseq____&logseq____<0\\\\\\\\\\n|x-3|logseq____&=-(x-3)=-x+3\\\\\\\\\\n-x-5logseq____&logseq____< 2(-x+3)\\\\\\\\\\n-x-5logseq____&logseq____<-2x+6 logseq____&|logseq____&+5+2x\\\\\\\\\\nxlogseq____<11\\\\\\\\\\n\\\\end{align}\\n\\\\begin{align}\\nL_1=\\\\left\\\\{x\\\\in\\\\Reals\\\\Big|xlogseq____<-5,x\\\\le11\\\\right\\\\}=(-\\\\infty,-5)\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[151,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[151,logseq____"^Flogseq____",155,536870923]],[logseq____"^15logseq____",[151,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[151,logseq____"^Vlogseq____",155,536870923]],[logseq____"^15logseq____",[151,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[151,logseq____"^Ulogseq____",112,536870923]],[logseq____"^15logseq____",[151,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[151,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[151,logseq____"^Ulogseq____",119,536870923]],[logseq____"^15logseq____",[151,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[151,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[151,logseq____"^;logseq____",logseq____"~u652fddf3-f078-46df-bafd-61dde65c8a82logseq____",536870923]],[logseq____"^15logseq____",[152,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 1}\\\\quad 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1a}\\\\quad x\\\\ge -2$\\n\\\\begin{align}\\n|x+2|logseq____&=x+2\\\\\\\\\\nx-3logseq____&\\\\le x+2 logseq____&|logseq____&+3-x\\\\\\\\\\n0logseq____&\\\\le5\\\\\\\\\\n\\\\end{align}\\n\\\\begin{align}\\nL_1=\\\\left\\\\{x\\\\in\\\\Reals\\\\Big|x\\\\ge3,x\\\\ge-2,0\\\\le5\\\\right\\\\}=\\\\left\\\\{x\\\\in\\\\Reals\\\\Big|x\\\\ge3\\\\right\\\\}=[3,\\\\infty)\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[153,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[153,logseq____"^Flogseq____",152,536870923]],[logseq____"^15logseq____",[153,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[153,logseq____"^Vlogseq____",150,536870923]],[logseq____"^15logseq____",[153,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[153,logseq____"^Ulogseq____",112,536870923]],[logseq____"^15logseq____",[153,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[153,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[153,logseq____"^Ulogseq____",119,536870923]],[logseq____"^15logseq____",[153,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[153,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[153,logseq____"^;logseq____",logseq____"~u652fddf3-0c5b-4c2d-8eba-7602a650037alogseq____",536870923]],[logseq____"^15logseq____",[154,logseq____"^Qlogseq____",logseq____"### [[Beispiele]]logseq____",536870923]],[logseq____"^15logseq____",[154,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[154,logseq____"^Flogseq____",141,536870923]],[logseq____"^15logseq____",[154,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[154,logseq____"^Vlogseq____",133,536870923]],[logseq____"^15logseq____",[154,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[154,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[154,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[154,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870923]],[logseq____"^15logseq____",[154,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[154,logseq____"^Hlogseq____",39,536870923]],[logseq____"^15logseq____",[154,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[154,logseq____"^;logseq____",logseq____"~u652fddf3-b996-4e5f-ba92-70f6622af32blogseq____",536870923]],[logseq____"^15logseq____",[155,logseq____"^Qlogseq____",logseq____"### Lösung mit [[Fallunterscheidung]]logseq____",536870923]],[logseq____"^15logseq____",[155,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[155,logseq____"^Flogseq____",134,536870923]],[logseq____"^15logseq____",[155,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[155,logseq____"^Vlogseq____",134,536870923]],[logseq____"^15logseq____",[155,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[155,logseq____"^Ulogseq____",112,536870923]],[logseq____"^15logseq____",[155,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[155,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[155,logseq____"^Ulogseq____",119,536870923]],[logseq____"^15logseq____",[155,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[155,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870923]],[logseq____"^15logseq____",[155,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[155,logseq____"^Hlogseq____",112,536870923]],[logseq____"^15logseq____",[155,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[155,logseq____"^;logseq____",logseq____"~u652fddf3-be4f-49b7-895a-0fe8c662ef10logseq____",536870923]],[logseq____"^15logseq____",[156,logseq____"^Qlogseq____",logseq____"logseq____",536870923]],[logseq____"^15logseq____",[156,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[156,logseq____"^Flogseq____",163,536870923]],[logseq____"^15logseq____",[156,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[156,logseq____"^Vlogseq____",163,536870923]],[logseq____"^15logseq____",[156,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[156,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[156,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[156,logseq____"^Ulogseq____",119,536870923]],[logseq____"^15logseq____",[156,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[156,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[156,logseq____"^;logseq____",logseq____"~u652fddf3-715c-4e0a-844e-13d6fe20e4d8logseq____",536870923]],[logseq____"^15logseq____",[157,logseq____"^Qlogseq____",logseq____"[[Beispiele]]\\n\\\\begin{align}\\n|5|=logseq____&5\\\\\\\\\\n|-3,3|logseq____&=3,3\\\\\\\\\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[157,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[157,logseq____"^Flogseq____",158,536870923]],[logseq____"^15logseq____",[157,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[157,logseq____"^Vlogseq____",162,536870923]],[logseq____"^15logseq____",[157,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[157,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[157,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[157,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[157,logseq____"^Hlogseq____",39,536870923]],[logseq____"^15logseq____",[157,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[157,logseq____"^;logseq____",logseq____"~u652fddf3-55b7-419f-9de2-e5de7366de5dlogseq____",536870923]],[logseq____"^15logseq____",[158,logseq____"^Qlogseq____",logseq____"Der [[Betrag]] ist der zu [[Abstand]] 0\\n\\\\begin{align}\\nf(x)=logseq____&|x|:=logseq____&\\\\begin{cases}\\nxlogseq____&x\\\\ge0\\\\\\\\\\n-xlogseq____&xlogseq____<0\\n\\\\end{cases}\\\\\\\\\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[158,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[158,logseq____"^Flogseq____",162,536870923]],[logseq____"^15logseq____",[158,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[158,logseq____"^Vlogseq____",162,536870923]],[logseq____"^15logseq____",[158,logseq____"^Ulogseq____",116,536870923]],[logseq____"^15logseq____",[158,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[158,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[158,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[158,logseq____"^Hlogseq____",116,536870923]],[logseq____"^15logseq____",[158,logseq____"^Hlogseq____",117,536870923]],[logseq____"^15logseq____",[158,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[158,logseq____"^;logseq____",logseq____"~u652fddf3-9b3d-4c80-8b01-8b0eb9c8bd4clogseq____",536870923]],[logseq____"^15logseq____",[159,logseq____"^Qlogseq____",logseq____"Kombinieren:\\n\\\\begin{align}\\nL=L_1\\\\cup L_2=(-2,8)\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[159,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[159,logseq____"^Flogseq____",160,536870923]],[logseq____"^15logseq____",[159,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[159,logseq____"^Vlogseq____",165,536870923]],[logseq____"^15logseq____",[159,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[159,logseq____"^Ulogseq____",112,536870923]],[logseq____"^15logseq____",[159,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[159,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[159,logseq____"^Ulogseq____",119,536870923]],[logseq____"^15logseq____",[159,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[159,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[159,logseq____"^;logseq____",logseq____"~u652fddf3-1c81-47e1-b443-a95e29a4de72logseq____",536870923]],[logseq____"^15logseq____",[160,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 2}\\\\quad x\\\\ge3$\\n\\\\begin{align}\\n|x-3|logseq____&=-x+3\\\\\\\\\\nx-3logseq____&logseq____<5 logseq____&|logseq____&+3\\\\\\\\\\nxlogseq____&logseq____<8\\\\\\\\\\nL_2logseq____&=[3,8)\\\\\\\\\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[160,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[160,logseq____"^Flogseq____",161,536870923]],[logseq____"^15logseq____",[160,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[160,logseq____"^Vlogseq____",165,536870923]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",112,536870923]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",119,536870923]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[160,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[160,logseq____"^;logseq____",logseq____"~u652fddf3-7cab-4a2b-acc6-27ac94ad2346logseq____",536870923]],[logseq____"^15logseq____",[161,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 1}\\\\quad xlogseq____<3$\\n\\\\begin{align}\\n|x-3|logseq____&=-x+3\\\\\\\\\\n-x+3logseq____&logseq____<5 logseq____&|logseq____&-3\\\\\\\\\\n-xlogseq____&logseq____<2 logseq____&|logseq____&*-1\\\\\\\\\\nxlogseq____&logseq____>-2\\\\\\\\\\nL_1logseq____&=(-2,3)\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[161,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[161,logseq____"^Flogseq____",165,536870923]],[logseq____"^15logseq____",[161,logseq____"^Xlogseq____",118,536870923]],[logseq____"^15logseq____",[161,logseq____"^Vlogseq____",165,536870923]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",39,536870923]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",112,536870923]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",117,536870923]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",118,536870923]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",119,536870923]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",122,536870923]],[logseq____"^15logseq____",[161,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[161,logseq____"^;logseq____",logseq____"~u652fddf3-609d-4d49-947a-7865ec0e46edlogseq____",536870923]],[logseq____"^15logseq____",[162,logseq____"^Qlogseq____",logseq____"## 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[[Aussageformel]]logseq____",536870925]],[logseq____"^15logseq____",[201,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[201,logseq____"^Flogseq____",210,536870925]],[logseq____"^15logseq____",[201,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[201,logseq____"^Vlogseq____",217,536870925]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",186,536870925]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[201,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870925]],[logseq____"^15logseq____",[201,logseq____"^Jlogseq____",[],536870925]],[logseq____"^15logseq____",[201,logseq____"^Hlogseq____",186,536870925]],[logseq____"^15logseq____",[201,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[201,logseq____"^;logseq____",logseq____"~u652fddf4-4096-417a-97e3-b2348cda15d4logseq____",536870925]],[logseq____"^15logseq____",[202,logseq____"^Qlogseq____",logseq____"### [[Kontradiktion]]logseq____",536870925]],[logseq____"^15logseq____",[202,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[202,logseq____"^Flogseq____",200,536870925]],[logseq____"^15logseq____",[202,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[202,logseq____"^Vlogseq____",217,536870925]],[logseq____"^15logseq____",[202,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[202,logseq____"^Ulogseq____",179,536870925]],[logseq____"^15logseq____",[202,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[202,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[202,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870925]],[logseq____"^15logseq____",[202,logseq____"^Jlogseq____",[],536870925]],[logseq____"^15logseq____",[202,logseq____"^Hlogseq____",179,536870925]],[logseq____"^15logseq____",[202,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[202,logseq____"^;logseq____",logseq____"~u652fddf4-ac33-4fd1-a1ed-8232b4877122logseq____",536870925]],[logseq____"^15logseq____",[203,logseq____"^Qlogseq____",logseq____"[Variable]([[Variablen]]), die den Wert $w$ oder $f$ annimmtlogseq____",536870925]],[logseq____"^15logseq____",[203,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[203,logseq____"^Flogseq____",210,536870925]],[logseq____"^15logseq____",[203,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[203,logseq____"^Vlogseq____",210,536870925]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",173,536870925]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",180,536870925]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[203,logseq____"^Hlogseq____",180,536870925]],[logseq____"^15logseq____",[203,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[203,logseq____"^;logseq____",logseq____"~u652fddf4-a91d-4680-b43c-c8f63f9003calogseq____",536870925]],[logseq____"^15logseq____",[204,logseq____"^Qlogseq____",logseq____"[[Wikipedia]]logseq____",536870925]],[logseq____"^15logseq____",[204,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[204,logseq____"^Flogseq____",191,536870925]],[logseq____"^15logseq____",[204,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[204,logseq____"^Vlogseq____",199,536870925]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",174,536870925]],[logseq____"^15logseq____",[204,logseq____"^Hlogseq____",174,536870925]],[logseq____"^15logseq____",[204,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[204,logseq____"^;logseq____",logseq____"~u652fddf4-d866-4a53-b484-ed48cfa4feablogseq____",536870925]],[logseq____"^15logseq____",[205,logseq____"^Qlogseq____",logseq____"Alternativ: $p\\\\equiv q$, falls $p\\\\leftrightarrow q$ [[Tantologie]]logseq____",536870925]],[logseq____"^15logseq____",[205,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[205,logseq____"^Flogseq____",194,536870925]],[logseq____"^15logseq____",[205,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[205,logseq____"^Vlogseq____",217,536870925]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",178,536870925]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[205,logseq____"^Hlogseq____",178,536870925]],[logseq____"^15logseq____",[205,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[205,logseq____"^;logseq____",logseq____"~u652fddf4-1dec-4f3a-885b-13c9104a4c66logseq____",536870925]],[logseq____"^15logseq____",[206,logseq____"^Qlogseq____",logseq____"Formel, die konstant $f$ istlogseq____",536870925]],[logseq____"^15logseq____",[206,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[206,logseq____"^Flogseq____",202,536870925]],[logseq____"^15logseq____",[206,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[206,logseq____"^Vlogseq____",202,536870925]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",179,536870925]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[206,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[206,logseq____"^;logseq____",logseq____"~u652fddf4-7f77-41fc-98de-681b9933ffcblogseq____",536870925]],[logseq____"^15logseq____",[207,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(p\\\\rightarrow q)logseq____&\\\\equiv((\\\\neg q)\\\\rightarrow(\\\\neg p))\\\\\\\\\\n(p\\\\land(p\\\\rightarrow q)) logseq____&\\\\rightarrow q logseq____&\\\\quad \\\\text{Tantologie}\\n\\\\end{align}logseq____",536870925]],[logseq____"^15logseq____",[207,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[207,logseq____"^Flogseq____",195,536870925]],[logseq____"^15logseq____",[207,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[207,logseq____"^Vlogseq____",195,536870925]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",39,536870925]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[207,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[207,logseq____"^;logseq____",logseq____"~u652fddf4-12ab-4f98-a349-4d9f297b19delogseq____",536870925]],[logseq____"^15logseq____",[208,logseq____"^Qlogseq____",logseq____"Jede mögliche [[Belegung]] wird ein [[Wahrheitswert]] der [[Formel]] zugeordnet (nach [[Tabellenregeln]])logseq____",536870925]],[logseq____"^15logseq____",[208,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[208,logseq____"^Flogseq____",212,536870925]],[logseq____"^15logseq____",[208,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[208,logseq____"^Vlogseq____",212,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",172,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",175,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",176,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",183,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",186,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",190,536870925]],[logseq____"^15logseq____",[208,logseq____"^Hlogseq____",175,536870925]],[logseq____"^15logseq____",[208,logseq____"^Hlogseq____",176,536870925]],[logseq____"^15logseq____",[208,logseq____"^Hlogseq____",183,536870925]],[logseq____"^15logseq____",[208,logseq____"^Hlogseq____",190,536870925]],[logseq____"^15logseq____",[208,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[208,logseq____"^;logseq____",logseq____"~u652fddf4-3255-4859-8488-e35337db3ab6logseq____",536870925]],[logseq____"^15logseq____",[209,logseq____"^Qlogseq____",logseq____"Entsteht durch sukzessive [[Verknüpfungen]] wie oben an Variablen ergibt #FIXMElogseq____",536870925]],[logseq____"^15logseq____",[209,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[209,logseq____"^Flogseq____",201,536870925]],[logseq____"^15logseq____",[209,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[209,logseq____"^Vlogseq____",201,536870925]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",46,536870925]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",185,536870925]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",186,536870925]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[209,logseq____"^Hlogseq____",46,536870925]],[logseq____"^15logseq____",[209,logseq____"^Hlogseq____",185,536870925]],[logseq____"^15logseq____",[209,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[209,logseq____"^;logseq____",logseq____"~u652fddf4-8580-403a-8191-b214333567a8logseq____",536870925]],[logseq____"^15logseq____",[210,logseq____"^Qlogseq____",logseq____"### [[Aussagenlogische Variable]]logseq____",536870925]],[logseq____"^15logseq____",[210,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[210,logseq____"^Flogseq____",213,536870925]],[logseq____"^15logseq____",[210,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[210,logseq____"^Vlogseq____",217,536870925]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",173,536870925]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[210,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870925]],[logseq____"^15logseq____",[210,logseq____"^Jlogseq____",[],536870925]],[logseq____"^15logseq____",[210,logseq____"^Hlogseq____",173,536870925]],[logseq____"^15logseq____",[210,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[210,logseq____"^;logseq____",logseq____"~u652fddf4-d545-44e8-9c4c-38d22028e4fdlogseq____",536870925]],[logseq____"^15logseq____",[211,logseq____"^Qlogseq____",logseq____"Formel, die konstant $w$ istlogseq____",536870925]],[logseq____"^15logseq____",[211,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[211,logseq____"^Flogseq____",200,536870925]],[logseq____"^15logseq____",[211,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[211,logseq____"^Vlogseq____",200,536870925]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",178,536870925]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[211,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[211,logseq____"^;logseq____",logseq____"~u652fddf4-c179-40c7-8884-41aab6e7af98logseq____",536870925]],[logseq____"^15logseq____",[212,logseq____"^Qlogseq____",logseq____"### [[Wahrheitswerteverlauf]] einer [[Aussageformel]]logseq____",536870925]],[logseq____"^15logseq____",[212,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[212,logseq____"^Flogseq____",196,536870925]],[logseq____"^15logseq____",[212,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[212,logseq____"^Vlogseq____",217,536870925]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",172,536870925]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",186,536870925]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[212,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870925]],[logseq____"^15logseq____",[212,logseq____"^Jlogseq____",[],536870925]],[logseq____"^15logseq____",[212,logseq____"^Hlogseq____",172,536870925]],[logseq____"^15logseq____",[212,logseq____"^Hlogseq____",186,536870925]],[logseq____"^15logseq____",[212,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[212,logseq____"^;logseq____",logseq____"~u652fddf4-1e28-415b-ac73-6c42d9890faclogseq____",536870925]],[logseq____"^15logseq____",[213,logseq____"^Qlogseq____",logseq____"Soweit die \\logseq____"[[Syntax]]\\logseq____", jetzt die [[Semantik]]. Die [Wahrheitswerte]([[Wahrheitswert]])\\n ergeben sich aus den Wahrheitswerten von $p$,$q$ gemäß folgender Tabelle ([[Tabellenregeln]]).\\n|$p$|$q$||$p\\\\land q$|$p\\\\lor q$|$\\\\neg p$|$p\\\\rightarrow q$|$p\\\\leftrightarrow q$|\\n|---|----||-----------|---------|---------|-----------------|---------------------|\\n|f|f||f|f|w|w|w|\\n|f|w||f|w|w|w|f|\\n|w|f||f|w|f|f|f|\\n|w|w||w|w|f|w|w|logseq____",536870925]],[logseq____"^15logseq____",[213,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[213,logseq____"^Flogseq____",198,536870925]],[logseq____"^15logseq____",[213,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[213,logseq____"^Vlogseq____",217,536870925]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",176,536870925]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",181,536870925]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",183,536870925]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",187,536870925]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[213,logseq____"^Hlogseq____",176,536870925]],[logseq____"^15logseq____",[213,logseq____"^Hlogseq____",181,536870925]],[logseq____"^15logseq____",[213,logseq____"^Hlogseq____",183,536870925]],[logseq____"^15logseq____",[213,logseq____"^Hlogseq____",187,536870925]],[logseq____"^15logseq____",[213,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[213,logseq____"^;logseq____",logseq____"~u652fddf4-0ece-4f7a-b845-371c5657a7f4logseq____",536870925]],[logseq____"^15logseq____",[214,logseq____"^Qlogseq____",logseq____"# 1. [[Aussagen]]logseq____",536870925]],[logseq____"^15logseq____",[214,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[214,logseq____"^Flogseq____",199,536870925]],[logseq____"^15logseq____",[214,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[214,logseq____"^Vlogseq____",169,536870925]],[logseq____"^15logseq____",[214,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[214,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[214,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870925]],[logseq____"^15logseq____",[214,logseq____"^Jlogseq____",[],536870925]],[logseq____"^15logseq____",[214,logseq____"^Hlogseq____",184,536870925]],[logseq____"^15logseq____",[214,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[214,logseq____"^;logseq____",logseq____"~u652fddf4-bd0d-4062-8bf8-3201262f633blogseq____",536870925]],[logseq____"^15logseq____",[215,logseq____"^Qlogseq____",logseq____"Zuordnung von $w/f$ an jede [Variable]([[Variablen]]) einer [[Aussageformel]]logseq____",536870925]],[logseq____"^15logseq____",[215,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[215,logseq____"^Flogseq____",196,536870925]],[logseq____"^15logseq____",[215,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[215,logseq____"^Vlogseq____",196,536870925]],[logseq____"^15logseq____",[215,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[215,logseq____"^Ulogseq____",180,536870925]],[logseq____"^15logseq____",[215,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[215,logseq____"^Ulogseq____",186,536870925]],[logseq____"^15logseq____",[215,logseq____"^Ulogseq____",188,536870925]],[logseq____"^15logseq____",[215,logseq____"^Ulogseq____",190,536870925]],[logseq____"^15logseq____",[215,logseq____"^Hlogseq____",180,536870925]],[logseq____"^15logseq____",[215,logseq____"^Hlogseq____",186,536870925]],[logseq____"^15logseq____",[215,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[215,logseq____"^;logseq____",logseq____"~u652fddf4-977a-4881-a242-07394bbf65fblogseq____",536870925]],[logseq____"^15logseq____",[216,logseq____"^Qlogseq____",logseq____"Satz, der wahr oder falsch ist, d.h. der [[Wahrheitswert]] $w$ bzw $f$ hat ($t/f$, $1/0$, $\\\\text{wahr}/\\\\text{falsch}$)logseq____",536870925]],[logseq____"^15logseq____",[216,logseq____"^Ologseq____",logseq____"^16logseq____",536870925]],[logseq____"^15logseq____",[216,logseq____"^Flogseq____",214,536870925]],[logseq____"^15logseq____",[216,logseq____"^Xlogseq____",169,536870925]],[logseq____"^15logseq____",[216,logseq____"^Vlogseq____",214,536870925]],[logseq____"^15logseq____",[216,logseq____"^Ulogseq____",169,536870925]],[logseq____"^15logseq____",[216,logseq____"^Ulogseq____",176,536870925]],[logseq____"^15logseq____",[216,logseq____"^Ulogseq____",184,536870925]],[logseq____"^15logseq____",[216,logseq____"^Hlogseq____",176,536870925]],[logseq____"^15logseq____",[216,logseq____"^17logseq____",true,536870925]],[logseq____"^15logseq____",[216,logseq____"^;logseq____",logseq____"~u652fddf4-928f-4ef0-9d7c-2fb65c4bc216logseq____",536870925]],[logseq____"^15logseq____",[217,logseq____"^Qlogseq____",logseq____"## [[Verknüpfung von 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U}\\\\end{cases}$logseq____",536870926]],[logseq____"^15logseq____",[245,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[245,logseq____"^Flogseq____",256,536870926]],[logseq____"^15logseq____",[245,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[245,logseq____"^Vlogseq____",256,536870926]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",176,536870926]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",184,536870926]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[245,logseq____"^Hlogseq____",176,536870926]],[logseq____"^15logseq____",[245,logseq____"^Hlogseq____",184,536870926]],[logseq____"^15logseq____",[245,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[245,logseq____"^;logseq____",logseq____"~u652fddf4-5fe8-4c09-ba7e-b22fb52fa49blogseq____",536870926]],[logseq____"^15logseq____",[246,logseq____"^Qlogseq____",logseq____"Sei $x$ aus $\\\\Reals$ beliebig. Setze $y:=x+1$ (ist aus $\\\\Reals$). Dann ist $xlogseq____<x+1=y$ also $xlogseq____<y$. $\\\\blacksquare$logseq____",536870926]],[logseq____"^15logseq____",[246,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[246,logseq____"^Flogseq____",270,536870926]],[logseq____"^15logseq____",[246,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[246,logseq____"^Vlogseq____",270,536870926]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",39,536870926]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",228,536870926]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",231,536870926]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",232,536870926]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[246,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[246,logseq____"^;logseq____",logseq____"~u652fddf4-0879-45df-a607-fd6673184886logseq____",536870926]],[logseq____"^15logseq____",[247,logseq____"^Qlogseq____",logseq____"Verschiedene [[Quantoren]] dürfen **nicht** vertauscht werdenlogseq____",536870926]],[logseq____"^15logseq____",[247,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[247,logseq____"^Flogseq____",278,536870926]],[logseq____"^15logseq____",[247,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[247,logseq____"^Vlogseq____",260,536870926]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",228,536870926]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",232,536870926]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[247,logseq____"^Hlogseq____",232,536870926]],[logseq____"^15logseq____",[247,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[247,logseq____"^;logseq____",logseq____"~u652fddf4-93f7-45de-a875-726d0c79edc0logseq____",536870926]],[logseq____"^15logseq____",[248,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\logseq____"\\\\text{x ist prim}\\logseq____" \\\\quad logseq____& \\\\text{ist AF mit Variablen x über den natürlichen Zahlen}\\\\\\\\\\n\\logseq____"\\\\underbrace{x}_{x_1}logseq____<\\\\underbrace{y}_{x_2}\\logseq____"\\\\quad logseq____&\\\\text{ist AF mit Variablen }x,y\\\\text{ über die reellen Zahlen } \\\\underbrace{\\\\Reals}_{U_1},\\\\underbrace{\\\\Reals}_{U_2}\\\\\\\\\\n\\logseq____"\\\\text{x ist wahr}\\logseq____"\\\\quadlogseq____&\\\\text{ist AF mit Variablen x über dem Universum der Aussagen}\\n\\\\end{align}logseq____",536870926]],[logseq____"^15logseq____",[248,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[248,logseq____"^Flogseq____",274,536870926]],[logseq____"^15logseq____",[248,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[248,logseq____"^Vlogseq____",280,536870926]],[logseq____"^15logseq____",[248,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[248,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[248,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[248,logseq____"^;logseq____",logseq____"~u652fddf4-a80a-4348-818f-c3d6c9be4d79logseq____",536870926]],[logseq____"^15logseq____",[249,logseq____"^Qlogseq____",logseq____"$\\\\exists x:p(x)$ ist die Aussage: \\logseq____"Es gibt (wenigstens) ein x aus U für das $p(x)$ wahr ist\\nIhr [[Wahrheitswert]] ist $\\\\begin{cases}w \\\\quadlogseq____&\\\\text{falls }p(x)\\\\text{ w ist für (wenigstens) ein x aus U}\\\\\\\\f\\\\quad logseq____&\\\\text{falls }p(x)\\\\text{ f ist für alle x aus U}\\\\end{cases}$logseq____",536870926]],[logseq____"^15logseq____",[249,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[249,logseq____"^Flogseq____",273,536870926]],[logseq____"^15logseq____",[249,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[249,logseq____"^Vlogseq____",280,536870926]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",176,536870926]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[249,logseq____"^Hlogseq____",176,536870926]],[logseq____"^15logseq____",[249,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[249,logseq____"^;logseq____",logseq____"~u652fddf4-e595-4a74-ac57-e26ec9f26ea0logseq____",536870926]],[logseq____"^15logseq____",[250,logseq____"^Qlogseq____",logseq____"[[Aussageform]] (AF) über [Universen]([[Universen (Logik)]]) $U_1,\\\\cdots,U_n$:logseq____",536870926]],[logseq____"^15logseq____",[250,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[250,logseq____"^Flogseq____",280,536870926]],[logseq____"^15logseq____",[250,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[250,logseq____"^Vlogseq____",280,536870926]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",242,536870926]],[logseq____"^15logseq____",[250,logseq____"^Hlogseq____",238,536870926]],[logseq____"^15logseq____",[250,logseq____"^Hlogseq____",242,536870926]],[logseq____"^15logseq____",[250,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[250,logseq____"^;logseq____",logseq____"~u652fddf4-3bfb-42d5-b0b1-177b51947fc1logseq____",536870926]],[logseq____"^15logseq____",[251,logseq____"^Qlogseq____",logseq____"Sei $p(x,y)$ AF in Variablen $x,y$ über $U_1,U_2$\\n\\\\begin{align}\\n\\\\underbrace{\\\\forall x: \\\\underbrace{\\\\exists y: p(x,y)}_{\\\\text{AF in den Variablen }x\\\\text{ über }U_1}}_{\\\\text{Aussage}}\\n\\\\end{align}logseq____",536870926]],[logseq____"^15logseq____",[251,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[251,logseq____"^Flogseq____",253,536870926]],[logseq____"^15logseq____",[251,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[251,logseq____"^Vlogseq____",253,536870926]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",39,536870926]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",220,536870926]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[251,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[251,logseq____"^;logseq____",logseq____"~u652fddf4-e44e-4bd9-9d41-0b2c6ffbb834logseq____",536870926]],[logseq____"^15logseq____",[252,logseq____"^Qlogseq____",logseq____"[[Satz]] mit [[Variablen]] $x_1,\\\\cdots,x_n$, der bei Ersetzung jedes $x_j$ durch ein [Objekt]([[Objekt (Logik)]]) aus $U_j$ stets zu einer [Aussage]([[Aussagen]]) wirdlogseq____",536870926]],[logseq____"^15logseq____",[252,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[252,logseq____"^Flogseq____",250,536870926]],[logseq____"^15logseq____",[252,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[252,logseq____"^Vlogseq____",280,536870926]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",180,536870926]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",184,536870926]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",241,536870926]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",244,536870926]],[logseq____"^15logseq____",[252,logseq____"^Hlogseq____",180,536870926]],[logseq____"^15logseq____",[252,logseq____"^Hlogseq____",184,536870926]],[logseq____"^15logseq____",[252,logseq____"^Hlogseq____",241,536870926]],[logseq____"^15logseq____",[252,logseq____"^Hlogseq____",244,536870926]],[logseq____"^15logseq____",[252,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[252,logseq____"^;logseq____",logseq____"~u652fddf4-6e3a-494b-814f-86923990fa9alogseq____",536870926]],[logseq____"^15logseq____",[253,logseq____"^Qlogseq____",logseq____"Schachtelung von quantoren ([Beispiel]([[Beispiele]])):logseq____",536870926]],[logseq____"^15logseq____",[253,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[253,logseq____"^Flogseq____",266,536870926]],[logseq____"^15logseq____",[253,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[253,logseq____"^Vlogseq____",272,536870926]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",39,536870926]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",220,536870926]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[253,logseq____"^Hlogseq____",39,536870926]],[logseq____"^15logseq____",[253,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[253,logseq____"^;logseq____",logseq____"~u652fddf4-c556-4719-bacb-4fe8aeaa3192logseq____",536870926]],[logseq____"^15logseq____",[254,logseq____"^Qlogseq____",logseq____"$\\\\blacksquare$ = Beweisende, auch [q.e.d.]([[Q.E.D.]]), [[w.z.b.w.]], QED, logseq____",536870926]],[logseq____"^15logseq____",[254,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[254,logseq____"^Flogseq____",246,536870926]],[logseq____"^15logseq____",[254,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[254,logseq____"^Vlogseq____",246,536870926]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",39,536870926]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",222,536870926]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",225,536870926]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",228,536870926]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",231,536870926]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",232,536870926]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[254,logseq____"^Hlogseq____",222,536870926]],[logseq____"^15logseq____",[254,logseq____"^Hlogseq____",225,536870926]],[logseq____"^15logseq____",[254,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[254,logseq____"^;logseq____",logseq____"~u652fddf4-3898-44c3-bcd3-b22f95764647logseq____",536870926]],[logseq____"^15logseq____",[255,logseq____"^Qlogseq____",logseq____"see: [[assoziativ]] [[kommutativ]] [[idenpotent]] [[wechselseitig distributiv]] [[distributiv]] [[de Morgansche Regeln]]logseq____",536870926]],[logseq____"^15logseq____",[255,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[255,logseq____"^Flogseq____",265,536870926]],[logseq____"^15logseq____",[255,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[255,logseq____"^Vlogseq____",265,536870926]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",221,536870926]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",226,536870926]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",229,536870926]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",230,536870926]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",236,536870926]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",237,536870926]],[logseq____"^15logseq____",[255,logseq____"^Hlogseq____",221,536870926]],[logseq____"^15logseq____",[255,logseq____"^Hlogseq____",226,536870926]],[logseq____"^15logseq____",[255,logseq____"^Hlogseq____",229,536870926]],[logseq____"^15logseq____",[255,logseq____"^Hlogseq____",230,536870926]],[logseq____"^15logseq____",[255,logseq____"^Hlogseq____",236,536870926]],[logseq____"^15logseq____",[255,logseq____"^Hlogseq____",237,536870926]],[logseq____"^15logseq____",[255,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[255,logseq____"^;logseq____",logseq____"~u652fddf4-684c-405a-93fa-2aeb9b632562logseq____",536870926]],[logseq____"^15logseq____",[256,logseq____"^Qlogseq____",logseq____"Sei $p(x)$ AF in der Variablen $x$ über $U$:logseq____",536870926]],[logseq____"^15logseq____",[256,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[256,logseq____"^Flogseq____",248,536870926]],[logseq____"^15logseq____",[256,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[256,logseq____"^Vlogseq____",280,536870926]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[256,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[256,logseq____"^;logseq____",logseq____"~u652fddf4-d0fa-4fd5-9e45-3e85457e99felogseq____",536870926]],[logseq____"^15logseq____",[257,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870926]],[logseq____"^15logseq____",[257,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[257,logseq____"^Flogseq____",269,536870926]],[logseq____"^15logseq____",[257,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[257,logseq____"^Vlogseq____",280,536870926]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",39,536870926]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[257,logseq____"^Hlogseq____",39,536870926]],[logseq____"^15logseq____",[257,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[257,logseq____"^;logseq____",logseq____"~u652fddf4-0ff3-408c-b2a2-3f0f6d99c876logseq____",536870926]],[logseq____"^15logseq____",[258,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nlogseq____&\\\\neg\\\\left(\\\\forall x :\\\\exists y :\\\\forall w :\\\\forall u :\\\\exists z:p(x,y,z)\\\\rightarrow q(u,w)\\\\right)\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:(\\\\neg(\\\\exists y: \\\\forall w: \\\\forall u: \\\\exists z: p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:\\\\forall y:(\\\\neg(\\\\forall w: \\\\forall u:\\\\exists z: p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:\\\\forall y:\\\\exists w:(\\\\neg(\\\\forall u:\\\\exists z:p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x: \\\\forall y: \\\\exists w: \\\\exists u : \\\\forall z:\\\\neg(p(x,y,z)\\\\rightarrow q(u,w))\\n\\\\end{align}logseq____",536870926]],[logseq____"^15logseq____",[258,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[258,logseq____"^Flogseq____",257,536870926]],[logseq____"^15logseq____",[258,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[258,logseq____"^Vlogseq____",257,536870926]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",39,536870926]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[258,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[258,logseq____"^;logseq____",logseq____"~u652fddf4-3978-49ac-84d7-bdc825991dadlogseq____",536870926]],[logseq____"^15logseq____",[259,logseq____"^Qlogseq____",logseq____"# [[Regeln]]logseq____",536870926]],[logseq____"^15logseq____",[259,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[259,logseq____"^Flogseq____",227,536870926]],[logseq____"^15logseq____",[259,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[259,logseq____"^Vlogseq____",227,536870926]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",220,536870926]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[259,logseq____"^?logseq____",[logseq____"^ 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x_n:p(x_1,x_2,\\\\cdots,x_n)\\\\notag\\\\\\\\\\nlogseq____&\\\\equiv\\\\forall_{x_{\\\\pi(1)}}:\\\\forall_{x_{\\\\pi(1)}}:\\\\cdots:\\\\forall_{x_{\\\\pi(1)}}:p(x_1,x_2,\\\\cdots,x_n)\\n\\\\end{align}logseq____",536870926]],[logseq____"^15logseq____",[260,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[260,logseq____"^Flogseq____",269,536870926]],[logseq____"^15logseq____",[260,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[260,logseq____"^Vlogseq____",269,536870926]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",228,536870926]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",232,536870926]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[260,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[260,logseq____"^;logseq____",logseq____"~u652fddf4-9a48-4983-8427-0a5866c1035blogseq____",536870926]],[logseq____"^15logseq____",[261,logseq____"^Qlogseq____",logseq____"Widerspruch = ↯, 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[[Allquantor]] 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$\\logseq____"xlogseq____<y\\logseq____"$ als AF über $\\\\Reals, 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[[Quantoren]] dürfen vertauscht 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das leere Universum gilt\\n\\\\begin{align}\\n\\\\forall x: p(x)\\\\quadlogseq____&Tautologie\\\\\\\\\\n\\\\exists x: p(x)\\\\quadlogseq____&Kontradiktion\\\\\\\\\\n\\\\end{align}logseq____",536870926]],[logseq____"^15logseq____",[271,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[271,logseq____"^Flogseq____",268,536870926]],[logseq____"^15logseq____",[271,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[271,logseq____"^Vlogseq____",280,536870926]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[271,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[271,logseq____"^;logseq____",logseq____"~u652fddf4-c5b1-43f5-a161-5884d61dff65logseq____",536870926]],[logseq____"^15logseq____",[272,logseq____"^Qlogseq____",logseq____"## 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Ersetzung von $x$ in der AF $\\logseq____"xlogseq____<y\\logseq____"$ durch eine konkrete Zahl entsteht eine AF in der Variablen $y$, z. Bsp. $\\logseq____"17logseq____<y\\logseq____"$logseq____",536870926]],[logseq____"^15logseq____",[275,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[275,logseq____"^Flogseq____",248,536870926]],[logseq____"^15logseq____",[275,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[275,logseq____"^Vlogseq____",248,536870926]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[275,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[275,logseq____"^;logseq____",logseq____"~u652fddf4-0a6a-48b6-ab5e-adad12a58bfflogseq____",536870926]],[logseq____"^15logseq____",[276,logseq____"^Qlogseq____",logseq____"Wäre $y$ aus $\\\\Reals$ mit $xlogseq____<y$ für alle $x$ aus $\\\\Reals$, so insbesondere auch $ylogseq____<y$, Widerspruch. $\\\\blacksquare$logseq____",536870926]],[logseq____"^15logseq____",[276,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[276,logseq____"^Flogseq____",279,536870926]],[logseq____"^15logseq____",[276,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[276,logseq____"^Vlogseq____",279,536870926]],[logseq____"^15logseq____",[276,logseq____"^Ulogseq____",39,536870926]],[logseq____"^15logseq____",[276,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[276,logseq____"^Ulogseq____",228,536870926]],[logseq____"^15logseq____",[276,logseq____"^Ulogseq____",231,536870926]],[logseq____"^15logseq____",[276,logseq____"^Ulogseq____",232,536870926]],[logseq____"^15logseq____",[276,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[276,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[276,logseq____"^;logseq____",logseq____"~u652fddf4-586a-4b73-bb22-f2e752942185logseq____",536870926]],[logseq____"^15logseq____",[277,logseq____"^Qlogseq____",logseq____"see: [[Große Azzoziativgesetze]] [[Große de Morgansche Regeln]] [[Negationsregeln]]logseq____",536870926]],[logseq____"^15logseq____",[277,logseq____"^Ologseq____",logseq____"^16logseq____",536870926]],[logseq____"^15logseq____",[277,logseq____"^Flogseq____",266,536870926]],[logseq____"^15logseq____",[277,logseq____"^Xlogseq____",227,536870926]],[logseq____"^15logseq____",[277,logseq____"^Vlogseq____",266,536870926]],[logseq____"^15logseq____",[277,logseq____"^Ulogseq____",220,536870926]],[logseq____"^15logseq____",[277,logseq____"^Ulogseq____",223,536870926]],[logseq____"^15logseq____",[277,logseq____"^Ulogseq____",227,536870926]],[logseq____"^15logseq____",[277,logseq____"^Ulogseq____",235,536870926]],[logseq____"^15logseq____",[277,logseq____"^Ulogseq____",238,536870926]],[logseq____"^15logseq____",[277,logseq____"^Ulogseq____",240,536870926]],[logseq____"^15logseq____",[277,logseq____"^Hlogseq____",223,536870926]],[logseq____"^15logseq____",[277,logseq____"^Hlogseq____",235,536870926]],[logseq____"^15logseq____",[277,logseq____"^Hlogseq____",240,536870926]],[logseq____"^15logseq____",[277,logseq____"^17logseq____",true,536870926]],[logseq____"^15logseq____",[277,logseq____"^;logseq____",logseq____"~u652fddf4-c488-4432-ae19-ea1e391fff55logseq____",536870926]],[logseq____"^15logseq____",[278,logseq____"^Qlogseq____",logseq____"wobei $\\\\pi$ eine \\logseq____"[[Permutation]]\\logseq____" der Zahl $(1,2,\\\\cdots,n)$ 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x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist 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[[Q.E.D.]]logseq____",536870934]],[logseq____"^15logseq____",[328,logseq____"^Ologseq____",logseq____"^16logseq____",536870934]],[logseq____"^15logseq____",[328,logseq____"^Flogseq____",327,536870934]],[logseq____"^15logseq____",[328,logseq____"^Xlogseq____",225,536870934]],[logseq____"^15logseq____",[328,logseq____"^Vlogseq____",325,536870934]],[logseq____"^15logseq____",[328,logseq____"^Ulogseq____",222,536870934]],[logseq____"^15logseq____",[328,logseq____"^Ulogseq____",225,536870934]],[logseq____"^15logseq____",[328,logseq____"^Hlogseq____",222,536870934]],[logseq____"^15logseq____",[328,logseq____"^17logseq____",true,536870934]],[logseq____"^15logseq____",[328,logseq____"^;logseq____",logseq____"~u652fddf4-34b1-4737-ab22-785ae0ab958elogseq____",536870934]],[logseq____"^15logseq____",[330,logseq____"^Qlogseq____",logseq____"logseq____",536870935]],[logseq____"^15logseq____",[330,logseq____"^Ologseq____",logseq____"^16logseq____",536870935]],[logseq____"^15logseq____",[330,logseq____"^Flogseq____",27,536870935]],[logseq____"^15logseq____",[330,logseq____"^Xlogseq____",27,536870935]],[logseq____"^15logseq____",[330,logseq____"^Vlogseq____",27,536870935]],[logseq____"^15logseq____",[330,logseq____"^Ulogseq____",27,536870935]],[logseq____"^15logseq____",[330,logseq____"^17logseq____",true,536870935]],[logseq____"^15logseq____",[330,logseq____"^;logseq____",logseq____"~u652fddf4-bb0b-4abd-8c48-58d64f5c6bc2logseq____",536870935]],[logseq____"^15logseq____",[332,logseq____"^Qlogseq____",logseq____"[[Wolframalpha]]logseq____",536870936]],[logseq____"^15logseq____",[332,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[332,logseq____"^Flogseq____",386,536870936]],[logseq____"^15logseq____",[332,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[332,logseq____"^Vlogseq____",386,536870936]],[logseq____"^15logseq____",[332,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[332,logseq____"^Ulogseq____",48,536870936]],[logseq____"^15logseq____",[332,logseq____"^Ulogseq____",50,536870936]],[logseq____"^15logseq____",[332,logseq____"^Hlogseq____",50,536870936]],[logseq____"^15logseq____",[332,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[332,logseq____"^;logseq____",logseq____"~u652fddf4-ce76-4aaa-b9ad-762a117f9f08logseq____",536870936]],[logseq____"^15logseq____",[333,logseq____"^Qlogseq____",logseq____"\\\\begin{array}{c}\\n{0\\\\choose 0}\\n\\\\\\\\\\n{1\\\\choose 0} {1\\\\choose 1}\\n\\\\\\\\\\n{2\\\\choose 0} {2\\\\choose 1} {2\\\\choose 2}\\n\\\\\\\\\\n{3\\\\choose 0} {3\\\\choose 1} {3\\\\choose 2} {3\\\\choose 3}\\n\\\\\\\\\\n\\\\cdots\\n\\\\end{array}logseq____",536870936]],[logseq____"^15logseq____",[333,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[333,logseq____"^Flogseq____",344,536870936]],[logseq____"^15logseq____",[333,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[333,logseq____"^Vlogseq____",365,536870936]],[logseq____"^15logseq____",[333,logseq____"^Ulogseq____",32,536870936]],[logseq____"^15logseq____",[333,logseq____"^Ulogseq____",36,536870936]],[logseq____"^15logseq____",[333,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[333,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[333,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[333,logseq____"^;logseq____",logseq____"~u652fddf4-f582-4472-8f9a-2c35acd82937logseq____",536870936]],[logseq____"^15logseq____",[334,logseq____"^Qlogseq____",logseq____"# [[Wurzelgleichung]]logseq____",536870936]],[logseq____"^15logseq____",[334,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[334,logseq____"^Flogseq____",353,536870936]],[logseq____"^15logseq____",[334,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[334,logseq____"^Vlogseq____",354,536870936]],[logseq____"^15logseq____",[334,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[334,logseq____"^Ulogseq____",34,536870936]],[logseq____"^15logseq____",[334,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[334,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[334,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[334,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[334,logseq____"^Hlogseq____",34,536870936]],[logseq____"^15logseq____",[334,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[334,logseq____"^;logseq____",logseq____"~u652fddf4-dd9b-461d-bbd3-9a02a56b9f1elogseq____",536870936]],[logseq____"^15logseq____",[335,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870936]],[logseq____"^15logseq____",[335,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[335,logseq____"^Flogseq____",345,536870936]],[logseq____"^15logseq____",[335,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[335,logseq____"^Vlogseq____",376,536870936]],[logseq____"^15logseq____",[335,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[335,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[335,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[335,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[335,logseq____"^Hlogseq____",39,536870936]],[logseq____"^15logseq____",[335,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[335,logseq____"^;logseq____",logseq____"~u652fddf4-2c54-4714-9179-960cbee26b5blogseq____",536870936]],[logseq____"^15logseq____",[336,logseq____"^Qlogseq____",logseq____"cosa -logseq____> beliebige Variable\\ncubus -logseq____> hoch 3logseq____",536870936]],[logseq____"^15logseq____",[336,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[336,logseq____"^Flogseq____",363,536870936]],[logseq____"^15logseq____",[336,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[336,logseq____"^Vlogseq____",363,536870936]],[logseq____"^15logseq____",[336,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[336,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[336,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[336,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[336,logseq____"^;logseq____",logseq____"~u652fddf4-01e1-4fe9-861f-4eb5923c3e9blogseq____",536870936]],[logseq____"^15logseq____",[337,logseq____"^Qlogseq____",logseq____"$\\\\text{Wenn }alogseq____>0\\\\text{, dann}$\\n\\\\begin{align}\\n\\\\frac b{\\\\sqrt a} = \\\\frac b{\\\\sqrt a}*\\\\frac{\\\\sqrt a}{\\\\sqrt a} = \\\\frac {b\\\\sqrt a}a\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[337,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[337,logseq____"^Flogseq____",347,536870936]],[logseq____"^15logseq____",[337,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[337,logseq____"^Vlogseq____",347,536870936]],[logseq____"^15logseq____",[337,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[337,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[337,logseq____"^Ulogseq____",43,536870936]],[logseq____"^15logseq____",[337,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[337,logseq____"^Ulogseq____",49,536870936]],[logseq____"^15logseq____",[337,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[337,logseq____"^;logseq____",logseq____"~u652fddf4-5e2f-44d0-8a4f-4bb2be6231dclogseq____",536870936]],[logseq____"^15logseq____",[338,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Potenzen]]logseq____",536870936]],[logseq____"^15logseq____",[338,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[338,logseq____"^Flogseq____",386,536870936]],[logseq____"^15logseq____",[338,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[338,logseq____"^Vlogseq____",44,536870936]],[logseq____"^15logseq____",[338,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[338,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[338,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[338,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[338,logseq____"^Hlogseq____",31,536870936]],[logseq____"^15logseq____",[338,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[338,logseq____"^;logseq____",logseq____"~u652fddf4-234f-4e74-bd5f-7d01c4b126f9logseq____",536870936]],[logseq____"^15logseq____",[339,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\underbrace{\\\\frac6{2+\\\\sqrt2}*\\\\frac{2-\\\\sqrt2}{2-\\\\sqrt2}\\n= \\\\frac{6(2-\\\\sqrt2)}{2^2-(\\\\sqrt2)^2}}_\\\\text{3. binomische Formel}\\nlogseq____&=\\\\frac{6(2-\\\\sqrt2)}2=3(2-\\\\sqrt2)\\\\\\\\\\n\\\\frac{a-b}{\\\\sqrt a+\\\\sqrt b}logseq____&=\\\\frac{a-b}{\\\\sqrt a + \\\\sqrt b}\\\\frac{\\\\sqrt a-\\\\sqrt b}{\\\\sqrt a-\\\\sqrt b}logseq____&, a,b,logseq____>0\\\\notag\\\\\\\\\\n=\\\\frac{(a-b)(\\\\sqrt a-\\\\sqrt b)}{(a-b)}logseq____&=\\\\sqrt a\\\\sqrt b\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[339,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[339,logseq____"^Flogseq____",352,536870936]],[logseq____"^15logseq____",[339,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[339,logseq____"^Vlogseq____",352,536870936]],[logseq____"^15logseq____",[339,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[339,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[339,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[339,logseq____"^Ulogseq____",43,536870936]],[logseq____"^15logseq____",[339,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[339,logseq____"^Ulogseq____",49,536870936]],[logseq____"^15logseq____",[339,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[339,logseq____"^;logseq____",logseq____"~u652fddf4-6c01-4d07-9cab-b6382d65ff0blogseq____",536870936]],[logseq____"^15logseq____",[340,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nax^2+bx+clogseq____&=0logseq____&|:a\\\\quadlogseq____&a\\\\ne0\\\\\\\\\\nx^2+\\\\frac ba x + \\\\frac ca logseq____&= 0\\\\\\\\\\nplogseq____&=\\\\frac ba,logseq____&q = \\\\frac ca\\\\\\\\\\nx^2+px+qlogseq____&=0\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[340,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[340,logseq____"^Flogseq____",353,536870936]],[logseq____"^15logseq____",[340,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[340,logseq____"^Vlogseq____",353,536870936]],[logseq____"^15logseq____",[340,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[340,logseq____"^Ulogseq____",33,536870936]],[logseq____"^15logseq____",[340,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[340,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[340,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[340,logseq____"^;logseq____",logseq____"~u652fddf4-4036-4fc1-b5f2-2defdc257119logseq____",536870936]],[logseq____"^15logseq____",[341,logseq____"^Qlogseq____",logseq____"logseq____",536870936]],[logseq____"^15logseq____",[341,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[341,logseq____"^Flogseq____",364,536870936]],[logseq____"^15logseq____",[341,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[341,logseq____"^Vlogseq____",338,536870936]],[logseq____"^15logseq____",[341,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[341,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[341,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[341,logseq____"^;logseq____",logseq____"~u652fddf4-0859-41c5-a94e-7a46d49cde76logseq____",536870936]],[logseq____"^15logseq____",[342,logseq____"^Qlogseq____",logseq____"$\\\\text{Für } x=\\\\frac25 \\\\text{ und } y=\\\\frac37 \\\\text{ ist }\\\\frac xy$\\n\\\\begin{align}\\n\\\\frac xy = \\\\frac{\\\\frac25}{\\\\frac37} = \\\\frac{2*7}{3*5}=\\\\frac{14}{15}\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[342,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[342,logseq____"^Flogseq____",360,536870936]],[logseq____"^15logseq____",[342,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[342,logseq____"^Vlogseq____",360,536870936]],[logseq____"^15logseq____",[342,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[342,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[342,logseq____"^;logseq____",logseq____"~u652fddf4-4b41-4bc9-a159-63d7107bf8dblogseq____",536870936]],[logseq____"^15logseq____",[343,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Logarithmen]]logseq____",536870936]],[logseq____"^15logseq____",[343,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[343,logseq____"^Flogseq____",338,536870936]],[logseq____"^15logseq____",[343,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[343,logseq____"^Vlogseq____",44,536870936]],[logseq____"^15logseq____",[343,logseq____"^Ulogseq____",35,536870936]],[logseq____"^15logseq____",[343,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[343,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[343,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[343,logseq____"^Hlogseq____",35,536870936]],[logseq____"^15logseq____",[343,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[343,logseq____"^;logseq____",logseq____"~u652fddf4-537e-44c5-baa4-21d5e10288e6logseq____",536870936]],[logseq____"^15logseq____",[344,logseq____"^Qlogseq____",logseq____"Das [[Pascalsche Dreieck]] lässt sich auch mit den entsprechenden Binomialkoeffizienten darstellenlogseq____",536870936]],[logseq____"^15logseq____",[344,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[344,logseq____"^Flogseq____",365,536870936]],[logseq____"^15logseq____",[344,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[344,logseq____"^Vlogseq____",365,536870936]],[logseq____"^15logseq____",[344,logseq____"^Ulogseq____",32,536870936]],[logseq____"^15logseq____",[344,logseq____"^Ulogseq____",36,536870936]],[logseq____"^15logseq____",[344,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[344,logseq____"^Ulogseq____",42,536870936]],[logseq____"^15logseq____",[344,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[344,logseq____"^Hlogseq____",42,536870936]],[logseq____"^15logseq____",[344,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[344,logseq____"^;logseq____",logseq____"~u652fddf4-d53e-41bc-9da8-88f9c95943ealogseq____",536870936]],[logseq____"^15logseq____",[345,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\na^nlogseq____&=\\\\underbrace{a*a*\\\\cdots*n}_n logseq____& a,blogseq____>0\\\\\\\\\\na^2b^2logseq____&=(ab)^n\\\\\\\\\\n\\\\frac{a^n}{b^n}logseq____&=\\\\left(\\\\frac ab\\\\right)^n\\\\\\\\\\na^na^mlogseq____&=a^{n+m}\\\\\\\\\\n\\\\frac1{a^n}logseq____&=a^{-n}\\\\\\\\\\n(a^n)^mlogseq____&=a^{nm}\\\\\\\\\\na^0logseq____&=1\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[345,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[345,logseq____"^Flogseq____",376,536870936]],[logseq____"^15logseq____",[345,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[345,logseq____"^Vlogseq____",376,536870936]],[logseq____"^15logseq____",[345,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[345,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[345,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[345,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[345,logseq____"^;logseq____",logseq____"~u652fddf4-10b3-4086-832b-795642284854logseq____",536870936]],[logseq____"^15logseq____",[346,logseq____"^Qlogseq____",logseq____"#FIXME $(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5=2+20x+40x^2$ ist Falschlogseq____",536870936]],[logseq____"^15logseq____",[346,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[346,logseq____"^Flogseq____",387,536870936]],[logseq____"^15logseq____",[346,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[346,logseq____"^Vlogseq____",373,536870936]],[logseq____"^15logseq____",[346,logseq____"^Ulogseq____",32,536870936]],[logseq____"^15logseq____",[346,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[346,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[346,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[346,logseq____"^Ulogseq____",46,536870936]],[logseq____"^15logseq____",[346,logseq____"^Hlogseq____",46,536870936]],[logseq____"^15logseq____",[346,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[346,logseq____"^;logseq____",logseq____"~u652fddf4-c604-4f7a-a5ef-3440faaf83a6logseq____",536870936]],[logseq____"^15logseq____",[347,logseq____"^Qlogseq____",logseq____"## [[Rationalmachen]] des [[Nenner]]slogseq____",536870936]],[logseq____"^15logseq____",[347,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[347,logseq____"^Flogseq____",389,536870936]],[logseq____"^15logseq____",[347,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[347,logseq____"^Vlogseq____",349,536870936]],[logseq____"^15logseq____",[347,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[347,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[347,logseq____"^Ulogseq____",43,536870936]],[logseq____"^15logseq____",[347,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[347,logseq____"^Ulogseq____",49,536870936]],[logseq____"^15logseq____",[347,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870936]],[logseq____"^15logseq____",[347,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[347,logseq____"^Hlogseq____",43,536870936]],[logseq____"^15logseq____",[347,logseq____"^Hlogseq____",49,536870936]],[logseq____"^15logseq____",[347,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[347,logseq____"^;logseq____",logseq____"~u652fddf4-accd-4551-89e5-a55f198294d3logseq____",536870936]],[logseq____"^15logseq____",[348,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn ergibtlogseq____",536870936]],[logseq____"^15logseq____",[348,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[348,logseq____"^Flogseq____",357,536870936]],[logseq____"^15logseq____",[348,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[348,logseq____"^Vlogseq____",350,536870936]],[logseq____"^15logseq____",[348,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[348,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[348,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[348,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[348,logseq____"^Ulogseq____",46,536870936]],[logseq____"^15logseq____",[348,logseq____"^Hlogseq____",46,536870936]],[logseq____"^15logseq____",[348,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[348,logseq____"^;logseq____",logseq____"~u652fddf4-7fa8-4614-bd97-ca55848ffe90logseq____",536870936]],[logseq____"^15logseq____",[349,logseq____"^Qlogseq____",logseq____"# [[Wurzeln]]logseq____",536870936]],[logseq____"^15logseq____",[349,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[349,logseq____"^Flogseq____",376,536870936]],[logseq____"^15logseq____",[349,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[349,logseq____"^Vlogseq____",354,536870936]],[logseq____"^15logseq____",[349,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[349,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[349,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[349,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[349,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[349,logseq____"^Hlogseq____",41,536870936]],[logseq____"^15logseq____",[349,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[349,logseq____"^;logseq____",logseq____"~u652fddf4-7aca-48aa-830b-5abaae42cf88logseq____",536870936]],[logseq____"^15logseq____",[350,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870936]],[logseq____"^15logseq____",[350,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[350,logseq____"^Flogseq____",370,536870936]],[logseq____"^15logseq____",[350,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[350,logseq____"^Vlogseq____",349,536870936]],[logseq____"^15logseq____",[350,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[350,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[350,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[350,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[350,logseq____"^Hlogseq____",39,536870936]],[logseq____"^15logseq____",[350,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[350,logseq____"^;logseq____",logseq____"~u652fddf4-4885-4a35-a92c-13fcf11bd9a5logseq____",536870936]],[logseq____"^15logseq____",[351,logseq____"^Qlogseq____",logseq____"#FIXME $\\\\sin^{-2}x+cos^2y=1,r$ ist falschlogseq____",536870936]],[logseq____"^15logseq____",[351,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[351,logseq____"^Flogseq____",383,536870936]],[logseq____"^15logseq____",[351,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[351,logseq____"^Vlogseq____",383,536870936]],[logseq____"^15logseq____",[351,logseq____"^Ulogseq____",40,536870936]],[logseq____"^15logseq____",[351,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[351,logseq____"^Ulogseq____",46,536870936]],[logseq____"^15logseq____",[351,logseq____"^Hlogseq____",46,536870936]],[logseq____"^15logseq____",[351,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[351,logseq____"^;logseq____",logseq____"~u652fddf4-1f12-45f6-a818-b0b7a1d3cb2blogseq____",536870936]],[logseq____"^15logseq____",[352,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870936]],[logseq____"^15logseq____",[352,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[352,logseq____"^Flogseq____",337,536870936]],[logseq____"^15logseq____",[352,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[352,logseq____"^Vlogseq____",347,536870936]],[logseq____"^15logseq____",[352,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[352,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[352,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[352,logseq____"^Ulogseq____",43,536870936]],[logseq____"^15logseq____",[352,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[352,logseq____"^Ulogseq____",49,536870936]],[logseq____"^15logseq____",[352,logseq____"^Hlogseq____",39,536870936]],[logseq____"^15logseq____",[352,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[352,logseq____"^;logseq____",logseq____"~u652fddf4-3ff7-4bec-a470-4cdb117f99cflogseq____",536870936]],[logseq____"^15logseq____",[353,logseq____"^Qlogseq____",logseq____"# [[Quadratische Gleichungen]]logseq____",536870936]],[logseq____"^15logseq____",[353,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[353,logseq____"^Flogseq____",349,536870936]],[logseq____"^15logseq____",[353,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[353,logseq____"^Vlogseq____",354,536870936]],[logseq____"^15logseq____",[353,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[353,logseq____"^Ulogseq____",33,536870936]],[logseq____"^15logseq____",[353,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[353,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[353,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[353,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[353,logseq____"^Hlogseq____",33,536870936]],[logseq____"^15logseq____",[353,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[353,logseq____"^;logseq____",logseq____"~u652fddf4-16b5-4e78-a6c7-6fd2871af755logseq____",536870936]],[logseq____"^15logseq____",[354,logseq____"^Qlogseq____",logseq____"# [[Potenzen]] und [[Wurzeln]]logseq____",536870936]],[logseq____"^15logseq____",[354,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[354,logseq____"^Flogseq____",381,536870936]],[logseq____"^15logseq____",[354,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[354,logseq____"^Vlogseq____",44,536870936]],[logseq____"^15logseq____",[354,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[354,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[354,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[354,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[354,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[354,logseq____"^Hlogseq____",31,536870936]],[logseq____"^15logseq____",[354,logseq____"^Hlogseq____",41,536870936]],[logseq____"^15logseq____",[354,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[354,logseq____"^;logseq____",logseq____"~u652fddf4-06f2-4b16-9e99-a81d08f5ae48logseq____",536870936]],[logseq____"^15logseq____",[355,logseq____"^Qlogseq____",logseq____"$\\\\text{Allgemein}$\\n\\\\begin{align}\\n(a+b)^2logseq____&=a^2+2ab+b^2logseq____&,\\\\text{ 1. Binomische Formel}\\\\\\\\\\n(a-b)^2logseq____&=a^2-2ab+b^2logseq____&,\\\\text{ 2. Binomische Formel}\\\\\\\\\\n(a+b)(a-b)logseq____&=a^2-b^2logseq____&,\\\\text{ 3. Binomische Formel}\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[355,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[355,logseq____"^Flogseq____",381,536870936]],[logseq____"^15logseq____",[355,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[355,logseq____"^Vlogseq____",381,536870936]],[logseq____"^15logseq____",[355,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[355,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[355,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[355,logseq____"^;logseq____",logseq____"~u652fddf4-b191-425b-8ff2-87487371189elogseq____",536870936]],[logseq____"^15logseq____",[356,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870936]],[logseq____"^15logseq____",[356,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[356,logseq____"^Flogseq____",377,536870936]],[logseq____"^15logseq____",[356,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[356,logseq____"^Vlogseq____",377,536870936]],[logseq____"^15logseq____",[356,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[356,logseq____"^Ulogseq____",33,536870936]],[logseq____"^15logseq____",[356,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[356,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[356,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[356,logseq____"^Ulogseq____",45,536870936]],[logseq____"^15logseq____",[356,logseq____"^Ulogseq____",47,536870936]],[logseq____"^15logseq____",[356,logseq____"^Hlogseq____",39,536870936]],[logseq____"^15logseq____",[356,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[356,logseq____"^;logseq____",logseq____"~u652fddf4-b560-4b94-9bf9-e88c0730d62dlogseq____",536870936]],[logseq____"^15logseq____",[357,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\sqrt{x^4+6x^2y+9y^2}logseq____&=\\\\sqrt{(x^2+3y)^2}=x^2+3y\\\\\\\\\\n\\\\sqrt {x + \\\\sqrt{x^2-y^2}}\\\\sqrt{x-\\\\sqrt{x^2-y^2}}logseq____&=\\\\sqrt{\\\\left(x + \\\\sqrt{x^2-y^2}\\\\right)\\\\left(x-\\\\sqrt{x^2-y^2}\\\\right)}\\\\notag\\\\\\\\\\nlogseq____&=\\\\sqrt{x^2-(x^2-y^2)}\\\\notag\\\\\\\\\\nlogseq____&=\\\\sqrt{y^2}=\\\\left|y\\\\right|\\\\\\\\\\n\\\\sqrt[5]{a\\\\sqrt[3]a}=\\\\sqrt[5]{\\\\sqrt[3]{a^3}\\\\sqrt[3]a}\\nlogseq____&=\\\\sqrt[5]{\\\\sqrt[3]{a^4}}\\n=\\\\sqrt[5]{a^\\\\frac43}\\n=a^{\\\\frac43\\\\frac15}=a^\\\\frac4{15}\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[357,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[357,logseq____"^Flogseq____",350,536870936]],[logseq____"^15logseq____",[357,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[357,logseq____"^Vlogseq____",350,536870936]],[logseq____"^15logseq____",[357,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[357,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[357,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[357,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[357,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[357,logseq____"^;logseq____",logseq____"~u652fddf4-5c6d-421a-a057-417aec9115fblogseq____",536870936]],[logseq____"^15logseq____",[358,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche Gleichungen sind richtig?}$\\n$\\\\text{Wenn }alogseq____>0\\\\text{, dann}$\\n\\\\begin{align}\\n\\\\frac{a^2}{\\\\sqrt{a}}logseq____&=a\\\\sqrt{a}logseq____&,r\\\\\\\\\\n\\\\frac{a^2}{\\\\sqrt{a}}logseq____&=a^2-\\\\sqrt{a}logseq____&,f\\\\\\\\\\n\\\\frac{a^2}{\\\\sqrt{a}}logseq____&=\\\\sqrt{a^3}logseq____&,r\\\\\\\\\\n\\\\frac{a^2}{\\\\sqrt{a}}logseq____&=a^{\\\\frac32}logseq____&,r\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[358,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[358,logseq____"^Flogseq____",371,536870936]],[logseq____"^15logseq____",[358,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[358,logseq____"^Vlogseq____",371,536870936]],[logseq____"^15logseq____",[358,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[358,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[358,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[358,logseq____"^;logseq____",logseq____"~u652fddf4-894a-47da-93ca-e2ae87d0ed70logseq____",536870936]],[logseq____"^15logseq____",[359,logseq____"^Qlogseq____",logseq____"https://www.wolframalpha.comlogseq____",536870936]],[logseq____"^15logseq____",[359,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[359,logseq____"^Flogseq____",332,536870936]],[logseq____"^15logseq____",[359,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[359,logseq____"^Vlogseq____",332,536870936]],[logseq____"^15logseq____",[359,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[359,logseq____"^Ulogseq____",48,536870936]],[logseq____"^15logseq____",[359,logseq____"^Ulogseq____",50,536870936]],[logseq____"^15logseq____",[359,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[359,logseq____"^;logseq____",logseq____"~u652fddf4-56fc-424d-b872-015479a9cadclogseq____",536870936]],[logseq____"^15logseq____",[360,logseq____"^Qlogseq____",logseq____"# Was ist Größer?logseq____",536870936]],[logseq____"^15logseq____",[360,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[360,logseq____"^Flogseq____",371,536870936]],[logseq____"^15logseq____",[360,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[360,logseq____"^Vlogseq____",44,536870936]],[logseq____"^15logseq____",[360,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[360,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[360,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[360,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[360,logseq____"^;logseq____",logseq____"~u652fddf4-4d4f-4de3-b446-bba1ef6156b2logseq____",536870936]],[logseq____"^15logseq____",[361,logseq____"^Qlogseq____",logseq____"$\\\\text{Häufige Falle:}$\\n\\\\begin{align}\\n\\\\left((-1)^2\\\\right)^\\\\frac12logseq____&=(-1)^\\\\frac22logseq____&=(-1)^1logseq____&=-1\\\\\\\\\\n\\\\left((-1)^2\\\\right)^\\\\frac12logseq____&=1^\\\\frac12logseq____&logseq____&=1\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[361,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[361,logseq____"^Flogseq____",358,536870936]],[logseq____"^15logseq____",[361,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[361,logseq____"^Vlogseq____",371,536870936]],[logseq____"^15logseq____",[361,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[361,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[361,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[361,logseq____"^;logseq____",logseq____"~u652fddf4-ac60-46ca-b46c-c709293bc283logseq____",536870936]],[logseq____"^15logseq____",[362,logseq____"^Qlogseq____",logseq____"Einsetzen, um zu überprüfen, ob man eine Fallunterscheidung brauchtlogseq____",536870936]],[logseq____"^15logseq____",[362,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[362,logseq____"^Flogseq____",378,536870936]],[logseq____"^15logseq____",[362,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[362,logseq____"^Vlogseq____",334,536870936]],[logseq____"^15logseq____",[362,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[362,logseq____"^Ulogseq____",34,536870936]],[logseq____"^15logseq____",[362,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[362,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[362,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[362,logseq____"^;logseq____",logseq____"~u652fddf4-6f21-4750-90bb-f2b4e7a45a49logseq____",536870936]],[logseq____"^15logseq____",[363,logseq____"^Qlogseq____",logseq____"$\\\\text{Alte Schreibweise}$\\n\\\\begin{align}\\n\\\\text{cosa }logseq____&\\\\text{plus }logseq____&\\\\text{cubus }logseq____&\\\\text{acq }logseq____&6\\\\\\\\\\nxlogseq____&+logseq____&x^3logseq____&=logseq____&6\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[363,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[363,logseq____"^Flogseq____",335,536870936]],[logseq____"^15logseq____",[363,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[363,logseq____"^Vlogseq____",376,536870936]],[logseq____"^15logseq____",[363,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[363,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[363,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[363,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[363,logseq____"^;logseq____",logseq____"~u652fddf4-1acc-48c8-943c-7911a16d3f6alogseq____",536870936]],[logseq____"^15logseq____",[364,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche }$[[Gleichungen]]$\\\\text{ sind richtig?}$\\n\\\\begin{align}\\ne^{x+y} logseq____&= e^x + e^y logseq____& ,f \\\\\\\\\\ne^{x+y} logseq____&= e^x * x^y logseq____& ,r \\\\\\\\\\ne^{x+y}logseq____&=e^{xy} logseq____& ,f\\\\\\\\\\ne^{ln(x+y)}logseq____&=x+y logseq____& ,x+ylogseq____>c \\\\\\\\\\ne^{ln(x+y)}logseq____&=e^x*e^ylogseq____&,f\\\\\\\\\\ne^{ln(x+y)}logseq____&=ln(e^x+y)logseq____&,x+ylogseq____>0\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[364,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[364,logseq____"^Flogseq____",338,536870936]],[logseq____"^15logseq____",[364,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[364,logseq____"^Vlogseq____",338,536870936]],[logseq____"^15logseq____",[364,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[364,logseq____"^Ulogseq____",37,536870936]],[logseq____"^15logseq____",[364,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[364,logseq____"^Hlogseq____",37,536870936]],[logseq____"^15logseq____",[364,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[364,logseq____"^;logseq____",logseq____"~u652fddf4-0ba2-477f-9e63-0d4b72b4f940logseq____",536870936]],[logseq____"^15logseq____",[365,logseq____"^Qlogseq____",logseq____"## [[Binomialkoeffizient]]logseq____",536870936]],[logseq____"^15logseq____",[365,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[365,logseq____"^Flogseq____",373,536870936]],[logseq____"^15logseq____",[365,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[365,logseq____"^Vlogseq____",366,536870936]],[logseq____"^15logseq____",[365,logseq____"^Ulogseq____",32,536870936]],[logseq____"^15logseq____",[365,logseq____"^Ulogseq____",36,536870936]],[logseq____"^15logseq____",[365,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[365,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[365,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870936]],[logseq____"^15logseq____",[365,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[365,logseq____"^Hlogseq____",36,536870936]],[logseq____"^15logseq____",[365,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[365,logseq____"^;logseq____",logseq____"~u652fddf4-6d80-4143-a613-328cf6113dd3logseq____",536870936]],[logseq____"^15logseq____",[366,logseq____"^Qlogseq____",logseq____"## [[Pascalsches Dreieck]]logseq____",536870936]],[logseq____"^15logseq____",[366,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[366,logseq____"^Flogseq____",375,536870936]],[logseq____"^15logseq____",[366,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[366,logseq____"^Vlogseq____",381,536870936]],[logseq____"^15logseq____",[366,logseq____"^Ulogseq____",32,536870936]],[logseq____"^15logseq____",[366,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[366,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[366,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870936]],[logseq____"^15logseq____",[366,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[366,logseq____"^Hlogseq____",32,536870936]],[logseq____"^15logseq____",[366,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[366,logseq____"^;logseq____",logseq____"~u652fddf4-4b5f-46e3-bdac-ea3853e6ddbalogseq____",536870936]],[logseq____"^15logseq____",[367,logseq____"^Qlogseq____",logseq____"## [[pq-Formel]]logseq____",536870936]],[logseq____"^15logseq____",[367,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[367,logseq____"^Flogseq____",340,536870936]],[logseq____"^15logseq____",[367,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[367,logseq____"^Vlogseq____",353,536870936]],[logseq____"^15logseq____",[367,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[367,logseq____"^Ulogseq____",33,536870936]],[logseq____"^15logseq____",[367,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[367,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[367,logseq____"^Ulogseq____",47,536870936]],[logseq____"^15logseq____",[367,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870936]],[logseq____"^15logseq____",[367,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[367,logseq____"^Hlogseq____",47,536870936]],[logseq____"^15logseq____",[367,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[367,logseq____"^;logseq____",logseq____"~u652fddf4-d031-4b84-b163-b77a1ac5dd94logseq____",536870936]],[logseq____"^15logseq____",[368,logseq____"^Qlogseq____",logseq____"\\\\begin{array}{c} (a+b)^0logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&1 \\\\\\\\\\n(a+b)^1logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&1 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^2logseq____&logseq____&logseq____&logseq____&logseq____&logseq____& 1 logseq____&logseq____& 2 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^3logseq____&logseq____&logseq____&logseq____&logseq____&1 logseq____&logseq____& 3 logseq____&logseq____& 3 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^4logseq____&logseq____&logseq____&logseq____& 1 logseq____&logseq____& 4 logseq____&logseq____& 6 logseq____&logseq____& 4 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^5logseq____&logseq____&logseq____& 1 logseq____&logseq____& 5 logseq____&logseq____& 10 logseq____&logseq____& 10 logseq____&logseq____& 5 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^6logseq____&logseq____& 1 logseq____&logseq____& 6 logseq____&logseq____& 15 logseq____&logseq____& 20 logseq____&logseq____& 15 logseq____&logseq____& 6 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^7logseq____&1 logseq____&logseq____& 7 logseq____&logseq____&21 logseq____&logseq____& 35 logseq____&logseq____& 35 logseq____&logseq____& \\\\cdots\\\\end{array}logseq____",536870936]],[logseq____"^15logseq____",[368,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[368,logseq____"^Flogseq____",366,536870936]],[logseq____"^15logseq____",[368,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[368,logseq____"^Vlogseq____",366,536870936]],[logseq____"^15logseq____",[368,logseq____"^Ulogseq____",32,536870936]],[logseq____"^15logseq____",[368,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[368,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[368,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[368,logseq____"^;logseq____",logseq____"~u652fddf4-eeb7-4f5a-80be-28e5c015e215logseq____",536870936]],[logseq____"^15logseq____",[369,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])\\n\\\\begin{align}\\nx^4-34x^2+225logseq____&=0\\\\\\\\\\n\\\\text{substituiert }ylogseq____&=x^2\\\\\\\\\\ny^2-34y+225logseq____&=0\\\\\\\\\\ny_{1,2}logseq____&=17\\\\pm\\\\sqrt{(-17)^2-225}\\\\\\\\\\nlogseq____&=17\\\\pm\\\\sqrt{289-225}\\nlogseq____&=17\\\\pm\\\\sqrt{64}=17\\\\pm8\\\\\\\\\\nlogseq____&=\\\\left.\\\\begin{cases}\\n25\\\\\\\\\\n9\\n\\\\end{cases}\\\\right|\\\\begin{array}{l}\\nx^2=25logseq____&\\\\rightarrow logseq____&x_1=5;logseq____&x_2=-5\\\\\\\\\\nx^2=9logseq____&\\\\rightarrow logseq____&x_1=3;logseq____&x_2=-3\\\\\\\\\\n\\\\end{array}\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[369,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[369,logseq____"^Flogseq____",377,536870936]],[logseq____"^15logseq____",[369,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[369,logseq____"^Vlogseq____",367,536870936]],[logseq____"^15logseq____",[369,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[369,logseq____"^Ulogseq____",33,536870936]],[logseq____"^15logseq____",[369,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[369,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[369,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[369,logseq____"^Ulogseq____",47,536870936]],[logseq____"^15logseq____",[369,logseq____"^Hlogseq____",39,536870936]],[logseq____"^15logseq____",[369,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[369,logseq____"^;logseq____",logseq____"~u652fddf4-d370-4a73-a502-075dcf720d1alogseq____",536870936]],[logseq____"^15logseq____",[370,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\sqrt[n]{a^n}logseq____&=1^\\\\frac nn=a^1=a\\\\\\\\\\n\\\\sqrt[n]{\\\\frac ab} logseq____&= \\\\frac{\\\\sqrt[n]a}{\\\\sqrt[n]b}\\\\\\\\\\n\\\\left(\\\\sqrt[n]a\\\\right)^mlogseq____&=a^\\\\frac mn\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[370,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[370,logseq____"^Flogseq____",384,536870936]],[logseq____"^15logseq____",[370,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[370,logseq____"^Vlogseq____",349,536870936]],[logseq____"^15logseq____",[370,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[370,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[370,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[370,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[370,logseq____"^;logseq____",logseq____"~u652fddf4-0999-440c-ad08-04041f4e1715logseq____",536870936]],[logseq____"^15logseq____",[371,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Wurzeln]]logseq____",536870936]],[logseq____"^15logseq____",[371,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[371,logseq____"^Flogseq____",343,536870936]],[logseq____"^15logseq____",[371,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[371,logseq____"^Vlogseq____",44,536870936]],[logseq____"^15logseq____",[371,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[371,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[371,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[371,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[371,logseq____"^Hlogseq____",41,536870936]],[logseq____"^15logseq____",[371,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[371,logseq____"^;logseq____",logseq____"~u652fddf4-b11e-486a-8dc0-fcd197f5d853logseq____",536870936]],[logseq____"^15logseq____",[372,logseq____"^Qlogseq____",logseq____"$\\\\text{Allgemein}$\\n\\\\begin{align}\\n\\\\ln a + \\\\ln b logseq____&= \\\\ln(a*b) \\\\\\\\\\n\\\\ln(-a) logseq____&= \\\\frac1{\\\\ln a}\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[372,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[372,logseq____"^Flogseq____",385,536870936]],[logseq____"^15logseq____",[372,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[372,logseq____"^Vlogseq____",343,536870936]],[logseq____"^15logseq____",[372,logseq____"^Ulogseq____",35,536870936]],[logseq____"^15logseq____",[372,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[372,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[372,logseq____"^;logseq____",logseq____"~u652fddf4-2d18-4f4d-bfc7-4b041a746cc1logseq____",536870936]],[logseq____"^15logseq____",[373,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870936]],[logseq____"^15logseq____",[373,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[373,logseq____"^Flogseq____",368,536870936]],[logseq____"^15logseq____",[373,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[373,logseq____"^Vlogseq____",366,536870936]],[logseq____"^15logseq____",[373,logseq____"^Ulogseq____",32,536870936]],[logseq____"^15logseq____",[373,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[373,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[373,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[373,logseq____"^Hlogseq____",39,536870936]],[logseq____"^15logseq____",[373,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[373,logseq____"^;logseq____",logseq____"~u652fddf4-a4fe-4881-aa92-ae4790b2589dlogseq____",536870936]],[logseq____"^15logseq____",[374,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Trigonometrie]]logseq____",536870936]],[logseq____"^15logseq____",[374,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[374,logseq____"^Flogseq____",360,536870936]],[logseq____"^15logseq____",[374,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[374,logseq____"^Vlogseq____",44,536870936]],[logseq____"^15logseq____",[374,logseq____"^Ulogseq____",40,536870936]],[logseq____"^15logseq____",[374,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[374,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[374,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[374,logseq____"^Hlogseq____",40,536870936]],[logseq____"^15logseq____",[374,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[374,logseq____"^;logseq____",logseq____"~u652fddf4-cec9-4ca9-b5ef-96a393788c6alogseq____",536870936]],[logseq____"^15logseq____",[375,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiel für die 3. Binomische Formel}$\\n\\\\begin{align}\\n\\\\frac{\\\\frac1{a-b}+\\\\frac1{a+b}}{\\\\frac1{a-b}-\\\\frac1{a+b}}\\nlogseq____&=\\\\frac{\\\\frac{a+b+a-b}{(a+b)*(a-b)}}{\\\\frac{a+b-(a-b)}{(a+b)*(a-b)}}\\nlogseq____&=\\\\frac{a+b+a-b}{a+b-a+b}\\nlogseq____&=\\\\frac{2a}{2b}\\nlogseq____&=\\\\frac ab\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[375,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[375,logseq____"^Flogseq____",355,536870936]],[logseq____"^15logseq____",[375,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[375,logseq____"^Vlogseq____",381,536870936]],[logseq____"^15logseq____",[375,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[375,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[375,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[375,logseq____"^;logseq____",logseq____"~u652fddf4-989b-45e7-af86-4d893c2d9f73logseq____",536870936]],[logseq____"^15logseq____",[376,logseq____"^Qlogseq____",logseq____"# [[Potenzen]]logseq____",536870936]],[logseq____"^15logseq____",[376,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[376,logseq____"^Flogseq____",354,536870936]],[logseq____"^15logseq____",[376,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[376,logseq____"^Vlogseq____",354,536870936]],[logseq____"^15logseq____",[376,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[376,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[376,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[376,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[376,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[376,logseq____"^Hlogseq____",31,536870936]],[logseq____"^15logseq____",[376,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[376,logseq____"^;logseq____",logseq____"~u652fddf4-932d-48dd-927e-79238c38dd99logseq____",536870936]],[logseq____"^15logseq____",[377,logseq____"^Qlogseq____",logseq____"[[Satz von Vieta]]\\n\\\\begin{align}\\nx^2+px+qlogseq____&=0 logseq____&logseq____& \\\\text{habe }x_{1,2}\\\\text{ als Lösung}\\\\\\\\\\nx_1+x_2logseq____&=-p logseq____&logseq____& x_1*x_2=q\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[377,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[377,logseq____"^Flogseq____",380,536870936]],[logseq____"^15logseq____",[377,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[377,logseq____"^Vlogseq____",367,536870936]],[logseq____"^15logseq____",[377,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[377,logseq____"^Ulogseq____",33,536870936]],[logseq____"^15logseq____",[377,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[377,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[377,logseq____"^Ulogseq____",45,536870936]],[logseq____"^15logseq____",[377,logseq____"^Ulogseq____",47,536870936]],[logseq____"^15logseq____",[377,logseq____"^Hlogseq____",45,536870936]],[logseq____"^15logseq____",[377,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[377,logseq____"^;logseq____",logseq____"~u652fddf4-27c5-4b19-beee-d6094dc70743logseq____",536870936]],[logseq____"^15logseq____",[378,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\sqrt{a+5}-\\\\sqrt{x+3}logseq____&=2\\\\sqrt(x+1) logseq____&|logseq____&()^2\\\\\\\\\\n\\\\left(\\\\sqrt{x+5}-\\\\sqrt{x+3}\\\\right)^2logseq____&=4(x+1)\\\\\\\\\\n(x+5)-2\\\\sqrt{x+5}\\\\sqrt{x+3}+(x+3)logseq____&=4x+4logseq____&|logseq____&-(2x+8)\\\\\\\\\\n-2\\\\sqrt{(x+5)(x+3)}logseq____&=2x-4 logseq____&|logseq____&()^2\\\\\\\\\\n4(x+5)(x+3)logseq____&=4x^2-16x+16\\\\\\\\\\n4x^2+32x+60logseq____&=4x^2-16x+16logseq____&|logseq____&-4x^2+16x-16\\\\\\\\\\n48x+44logseq____&=0\\\\quad x=-\\\\frac{11}{12}\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[378,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[378,logseq____"^Flogseq____",334,536870936]],[logseq____"^15logseq____",[378,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[378,logseq____"^Vlogseq____",334,536870936]],[logseq____"^15logseq____",[378,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[378,logseq____"^Ulogseq____",34,536870936]],[logseq____"^15logseq____",[378,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[378,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[378,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[378,logseq____"^;logseq____",logseq____"~u652fddf4-9c3c-4213-b231-00548bc08c76logseq____",536870936]],[logseq____"^15logseq____",[379,logseq____"^Qlogseq____",logseq____"$\\\\frac12 \\\\sqrt2, \\\\frac12, \\\\frac12\\\\sqrt3 \\\\text{ sind werte von }\\\\sin x$\\n\\\\begin{align}\\n\\\\sin 30\\\\degreelogseq____&=\\\\frac12\\\\\\\\\\n\\\\sin 45\\\\degree logseq____&= \\\\frac12\\\\sqrt2 \\\\\\\\\\n\\\\sin 60\\\\degree logseq____&= \\\\frac12 \\\\sqrt3 \\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[379,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[379,logseq____"^Flogseq____",374,536870936]],[logseq____"^15logseq____",[379,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[379,logseq____"^Vlogseq____",374,536870936]],[logseq____"^15logseq____",[379,logseq____"^Ulogseq____",40,536870936]],[logseq____"^15logseq____",[379,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[379,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[379,logseq____"^;logseq____",logseq____"~u652fddf4-878c-48cd-a5b2-c24d2a406e7blogseq____",536870936]],[logseq____"^15logseq____",[380,logseq____"^Qlogseq____",logseq____"$\\\\text{Herleitung mit }$[[Quadratische Ergänzung]]\\n\\\\begin{align}\\nx^2+px+qlogseq____&=0logseq____&|logseq____&-q\\\\\\\\\\nx^2+pxlogseq____&=-qlogseq____&|logseq____&+\\\\left(\\\\frac p2\\\\right)^2\\\\\\\\\\nx^2+px+\\\\left(\\\\frac p2\\\\right)^2logseq____&=-q+\\\\left(\\\\frac p2\\\\right)^2\\\\\\\\\\n\\\\underbrace{\\\\left(x+\\\\frac p2\\\\right)^2}_\\\\text{1. binomische Formel}logseq____&=-q+\\\\left(\\\\frac p2\\\\right)^2logseq____&|logseq____&\\\\sqrt{()}\\\\\\\\\\n\\\\left|x+\\\\frac p2\\\\right|logseq____&=\\\\sqrt{\\\\frac{p^2}4-q}logseq____&,logseq____&\\\\text{ falls }\\\\frac{p^2}4-q\\\\ge0\\\\\\\\\\nx+\\\\frac p2logseq____&=\\\\pm\\\\sqrt{\\\\frac{p^2}4-q}logseq____&|logseq____&-\\\\frac p2\\\\\\\\\\nxlogseq____&=-\\\\frac p2\\\\pm\\\\sqrt{\\\\frac{p^2}4-q}\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[380,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[380,logseq____"^Flogseq____",367,536870936]],[logseq____"^15logseq____",[380,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[380,logseq____"^Vlogseq____",367,536870936]],[logseq____"^15logseq____",[380,logseq____"^Ulogseq____",30,536870936]],[logseq____"^15logseq____",[380,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[380,logseq____"^Ulogseq____",33,536870936]],[logseq____"^15logseq____",[380,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[380,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[380,logseq____"^Ulogseq____",47,536870936]],[logseq____"^15logseq____",[380,logseq____"^Hlogseq____",30,536870936]],[logseq____"^15logseq____",[380,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[380,logseq____"^;logseq____",logseq____"~u652fddf4-8a31-42fd-bc09-c59baa97cb41logseq____",536870936]],[logseq____"^15logseq____",[381,logseq____"^Qlogseq____",logseq____"# [[Binomische Formeln]]logseq____",536870936]],[logseq____"^15logseq____",[381,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[381,logseq____"^Flogseq____",374,536870936]],[logseq____"^15logseq____",[381,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[381,logseq____"^Vlogseq____",44,536870936]],[logseq____"^15logseq____",[381,logseq____"^Ulogseq____",38,536870936]],[logseq____"^15logseq____",[381,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[381,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[381,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[381,logseq____"^Hlogseq____",38,536870936]],[logseq____"^15logseq____",[381,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[381,logseq____"^;logseq____",logseq____"~u652fddf4-9e31-4f85-adfc-71f8b8749f5flogseq____",536870936]],[logseq____"^15logseq____",[382,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche quadratische Gleichung hat die Lösung}$\\n$x_1=3\\\\quad x_2=4$\\n\\\\begin{align}\\nx_1+x_2 = 7 logseq____&= p logseq____&, x_1x_2=12=q\\\\\\\\\\nx^2+px+qlogseq____&=0\\\\\\\\\\nx^2-7x+12logseq____&=0\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[382,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[382,logseq____"^Flogseq____",356,536870936]],[logseq____"^15logseq____",[382,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[382,logseq____"^Vlogseq____",356,536870936]],[logseq____"^15logseq____",[382,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[382,logseq____"^Ulogseq____",33,536870936]],[logseq____"^15logseq____",[382,logseq____"^Ulogseq____",39,536870936]],[logseq____"^15logseq____",[382,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[382,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[382,logseq____"^Ulogseq____",45,536870936]],[logseq____"^15logseq____",[382,logseq____"^Ulogseq____",47,536870936]],[logseq____"^15logseq____",[382,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[382,logseq____"^;logseq____",logseq____"~u652fddf4-7997-4e09-be09-b97df194a965logseq____",536870936]],[logseq____"^15logseq____",[383,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche Gleichungen sind richtig?}$\\n\\\\begin{align}\\n\\\\tan x logseq____&= \\\\frac{\\\\sin x}{\\\\cos x} logseq____&, \\\\cos x \\\\ne 0\\\\\\\\\\n\\\\sin^{-2}x-\\\\cos^2xlogseq____&=1logseq____&,f\\\\\\\\\\n\\\\sin x = \\\\cos x logseq____&= 1 logseq____&, f\\\\\\\\\\n\\\\sin^{-2}x+cos^2ylogseq____&=1logseq____&,r\\\\\\\\\\n1+tan^2xlogseq____&=\\\\frac1{\\\\cos^2x}logseq____&,\\\\cos x\\\\ne0\\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[383,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[383,logseq____"^Flogseq____",379,536870936]],[logseq____"^15logseq____",[383,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[383,logseq____"^Vlogseq____",374,536870936]],[logseq____"^15logseq____",[383,logseq____"^Ulogseq____",40,536870936]],[logseq____"^15logseq____",[383,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[383,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[383,logseq____"^;logseq____",logseq____"~u652fddf4-5d60-4947-8781-9edb8c5646f5logseq____",536870936]],[logseq____"^15logseq____",[384,logseq____"^Qlogseq____",logseq____"$\\\\text{Die Lösung von } x^n-a=0 \\\\text{ ist } x=\\\\sqrt[n]a$ #FIXME ich glaube hier fehlt logseq____>0 constraintlogseq____",536870936]],[logseq____"^15logseq____",[384,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[384,logseq____"^Flogseq____",349,536870936]],[logseq____"^15logseq____",[384,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[384,logseq____"^Vlogseq____",349,536870936]],[logseq____"^15logseq____",[384,logseq____"^Ulogseq____",31,536870936]],[logseq____"^15logseq____",[384,logseq____"^Ulogseq____",41,536870936]],[logseq____"^15logseq____",[384,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[384,logseq____"^Ulogseq____",46,536870936]],[logseq____"^15logseq____",[384,logseq____"^Hlogseq____",46,536870936]],[logseq____"^15logseq____",[384,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[384,logseq____"^;logseq____",logseq____"~u652fddf4-0a7d-47b7-9a1c-ef7e636bcce5logseq____",536870936]],[logseq____"^15logseq____",[385,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche Gleichungen sind richtig?}$\\n$\\\\text{Wenn } 10^x=y \\\\text{, dann}$\\n\\\\begin{align}\\nxlogseq____&=\\\\lg y logseq____& ,r\\\\\\\\\\nxlogseq____&=\\\\log_y10logseq____&,f\\\\\\\\\\nxlogseq____&=\\\\log_{10}ylogseq____&,r \\\\\\\\\\n\\\\ln 4 - \\\\ln 2 logseq____&= \\\\ln 2 logseq____&, r \\\\\\\\\\n\\\\ln 4 - \\\\ln 2 logseq____&= \\\\ln 8 logseq____&, f \\\\\\\\\\n\\\\ln 4 + \\\\ln 2 logseq____&= \\\\ln 8 logseq____&, r \\\\\\\\\\n\\\\end{align}logseq____",536870936]],[logseq____"^15logseq____",[385,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[385,logseq____"^Flogseq____",343,536870936]],[logseq____"^15logseq____",[385,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[385,logseq____"^Vlogseq____",343,536870936]],[logseq____"^15logseq____",[385,logseq____"^Ulogseq____",35,536870936]],[logseq____"^15logseq____",[385,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[385,logseq____"^17logseq____",true,536870936]],[logseq____"^15logseq____",[385,logseq____"^;logseq____",logseq____"~u652fddf4-e291-41f7-b28d-130131cc5dd7logseq____",536870936]],[logseq____"^15logseq____",[386,logseq____"^Qlogseq____",logseq____"# [[Links]]logseq____",536870936]],[logseq____"^15logseq____",[386,logseq____"^Ologseq____",logseq____"^16logseq____",536870936]],[logseq____"^15logseq____",[386,logseq____"^Flogseq____",44,536870936]],[logseq____"^15logseq____",[386,logseq____"^Xlogseq____",44,536870936]],[logseq____"^15logseq____",[386,logseq____"^Vlogseq____",44,536870936]],[logseq____"^15logseq____",[386,logseq____"^Ulogseq____",44,536870936]],[logseq____"^15logseq____",[386,logseq____"^Ulogseq____",48,536870936]],[logseq____"^15logseq____",[386,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870936]],[logseq____"^15logseq____",[386,logseq____"^Jlogseq____",[],536870936]],[logseq____"^15logseq____",[386,logseq____"^Hlogseq____",48,536870936]],[logseq____"^15logseq____",[386,logseq____"^17logseq____",false,536870936]],[logseq____"^15logseq____",[386,logseq____"^;logseq____",logseq____"~u652fddf4-3764-4493-a57b-113d14b19196logseq____",536870936]],[logseq____"^15logseq____",[387,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(2x+y)^4logseq____&=(2x)^4+4*(2x)^3y+6*(2x)^2y^2+4*2xy^3+y^4\\\\notag\\\\\\\\\\nlogseq____&=16x^4+32x^3y+24x^2y^2+8xy^3+y^4\\\\\\\\\\n(1+\\\\sqrt x)^5*(1-\\\\sqrt 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von [Punkten]([[Punkte]]) $(x,y)$ in der [Ebene]([[Ebenen]]) die den gleichen [[Abstand]] r ([[Radius]]) zu einem festen [Punkt]([[Punkte]]) 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aller reellen Zahlen zwischen 1 und 4 ohne 1 und 4\\n\\\\begin{align}\\\\left\\\\{x\\\\in\\\\Reals\\\\middle|-1logseq____<xlogseq____<4\\\\right\\\\}=:\\\\underbrace{(-1,4)}_\\\\text{Offenes 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$\\\\rightarrow$ 5 ist abstand von x zu 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incompletelogseq____",536870937]],[logseq____"^15logseq____",[417,logseq____"^Ologseq____",logseq____"^16logseq____",536870937]],[logseq____"^15logseq____",[417,logseq____"^Flogseq____",404,536870937]],[logseq____"^15logseq____",[417,logseq____"^Xlogseq____",118,536870937]],[logseq____"^15logseq____",[417,logseq____"^Vlogseq____",118,536870937]],[logseq____"^15logseq____",[417,logseq____"^Ulogseq____",46,536870937]],[logseq____"^15logseq____",[417,logseq____"^Ulogseq____",118,536870937]],[logseq____"^15logseq____",[417,logseq____"^Hlogseq____",46,536870937]],[logseq____"^15logseq____",[417,logseq____"^17logseq____",true,536870937]],[logseq____"^15logseq____",[417,logseq____"^;logseq____",logseq____"~u652fddf4-353b-40b2-9d60-ae2732a330ddlogseq____",536870937]],[logseq____"^15logseq____",[418,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 2}\\\\quad x\\\\ge3$\\n\\\\begin{align}\\n|x-3|logseq____&=-x+3\\\\\\\\\\nx-3logseq____&logseq____<5 logseq____&|logseq____&+3\\\\\\\\\\nxlogseq____&logseq____<8\\\\\\\\\\nL_2logseq____&=[3,8)\\\\\\\\\\n\\\\end{align}logseq____",536870937]],[logseq____"^15logseq____",[418,logseq____"^Ologseq____",logseq____"^16logseq____",536870937]],[logseq____"^15logseq____",[418,logseq____"^Flogseq____",392,536870937]],[logseq____"^15logseq____",[418,logseq____"^Xlogseq____",118,536870937]],[logseq____"^15logseq____",[418,logseq____"^Vlogseq____",414,536870937]],[logseq____"^15logseq____",[418,logseq____"^Ulogseq____",39,536870937]],[logseq____"^15logseq____",[418,logseq____"^Ulogseq____",112,536870937]],[logseq____"^15logseq____",[418,logseq____"^Ulogseq____",117,536870937]],[logseq____"^15logseq____",[418,logseq____"^Ulogseq____",118,536870937]],[logseq____"^15logseq____",[418,logseq____"^Ulogseq____",119,536870937]],[logseq____"^15logseq____",[418,logseq____"^Ulogseq____",122,536870937]],[logseq____"^15logseq____",[418,logseq____"^17logseq____",true,536870937]],[logseq____"^15logseq____",[418,logseq____"^;logseq____",logseq____"~u652fddf4-a374-4a2f-a4eb-158a22527f08logseq____",536870937]],[logseq____"^15logseq____",[419,logseq____"^Qlogseq____",logseq____"Der [[Betrag]] ist der zu [[Abstand]] 0\\n\\\\begin{align}\\nf(x)=logseq____&|x|:=logseq____&\\\\begin{cases}\\nxlogseq____&x\\\\ge0\\\\\\\\\\n-xlogseq____&xlogseq____<0\\n\\\\end{cases}\\\\\\\\\\n\\\\end{align}logseq____",536870937]],[logseq____"^15logseq____",[419,logseq____"^Ologseq____",logseq____"^16logseq____",536870937]],[logseq____"^15logseq____",[419,logseq____"^Flogseq____",403,536870937]],[logseq____"^15logseq____",[419,logseq____"^Xlogseq____",118,536870937]],[logseq____"^15logseq____",[419,logseq____"^Vlogseq____",403,536870937]],[logseq____"^15logseq____",[419,logseq____"^Ulogseq____",116,536870937]],[logseq____"^15logseq____",[419,logseq____"^Ulogseq____",117,536870937]],[logseq____"^15logseq____",[419,logseq____"^Ulogseq____",118,536870937]],[logseq____"^15logseq____",[419,logseq____"^Ulogseq____",122,536870937]],[logseq____"^15logseq____",[419,logseq____"^Hlogseq____",116,536870937]],[logseq____"^15logseq____",[419,logseq____"^Hlogseq____",117,536870937]],[logseq____"^15logseq____",[419,logseq____"^17logseq____",true,536870937]],[logseq____"^15logseq____",[419,logseq____"^;logseq____",logseq____"~u652fddf4-823a-4b5a-9478-967f45200c4elogseq____",536870937]],[logseq____"^15logseq____",[420,logseq____"^Qlogseq____",logseq____"$$L=\\\\left\\\\{\\\\underbrace{x\\\\in\\\\Reals}_{0\\\\notin 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1b}\\\\quad xlogseq____<-2\\\\quad\\\\text{Wiederspricht Bedingung}\\\\quad x\\\\ge3$logseq____",536870937]],[logseq____"^15logseq____",[422,logseq____"^Ologseq____",logseq____"^16logseq____",536870937]],[logseq____"^15logseq____",[422,logseq____"^Flogseq____",415,536870937]],[logseq____"^15logseq____",[422,logseq____"^Xlogseq____",118,536870937]],[logseq____"^15logseq____",[422,logseq____"^Vlogseq____",396,536870937]],[logseq____"^15logseq____",[422,logseq____"^Ulogseq____",39,536870937]],[logseq____"^15logseq____",[422,logseq____"^Ulogseq____",112,536870937]],[logseq____"^15logseq____",[422,logseq____"^Ulogseq____",117,536870937]],[logseq____"^15logseq____",[422,logseq____"^Ulogseq____",118,536870937]],[logseq____"^15logseq____",[422,logseq____"^Ulogseq____",119,536870937]],[logseq____"^15logseq____",[422,logseq____"^Ulogseq____",122,536870937]],[logseq____"^15logseq____",[422,logseq____"^17logseq____",true,536870937]],[logseq____"^15logseq____",[422,logseq____"^;logseq____",logseq____"~u652fddf4-5f9b-4496-a008-99b2f0ec37b4logseq____",536870937]],[logseq____"^15logseq____",[423,logseq____"^Qlogseq____",logseq____"logseq____",536870937]],[logseq____"^15logseq____",[423,logseq____"^Ologseq____",logseq____"^16logseq____",536870937]],[logseq____"^15logseq____",[423,logseq____"^Flogseq____",416,536870937]],[logseq____"^15logseq____",[423,logseq____"^Xlogseq____",118,536870937]],[logseq____"^15logseq____",[423,logseq____"^Vlogseq____",416,536870937]],[logseq____"^15logseq____",[423,logseq____"^Ulogseq____",39,536870937]],[logseq____"^15logseq____",[423,logseq____"^Ulogseq____",117,536870937]],[logseq____"^15logseq____",[423,logseq____"^Ulogseq____",118,536870937]],[logseq____"^15logseq____",[423,logseq____"^Ulogseq____",119,536870937]],[logseq____"^15logseq____",[423,logseq____"^Ulogseq____",122,536870937]],[logseq____"^15logseq____",[423,logseq____"^17logseq____",true,536870937]],[logseq____"^15logseq____",[423,logseq____"^;logseq____",logseq____"~u652fddf4-8525-41a0-a029-4cf442d0db3blogseq____",536870937]],[logseq____"^15logseq____",[424,logseq____"^Qlogseq____",logseq____"Kombinieren:\\n\\\\begin{align}\\nL=L_1\\\\cup 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[[Beispiele]]logseq____",536870937]],[logseq____"^15logseq____",[425,logseq____"^Ologseq____",logseq____"^16logseq____",536870937]],[logseq____"^15logseq____",[425,logseq____"^Flogseq____",408,536870937]],[logseq____"^15logseq____",[425,logseq____"^Xlogseq____",118,536870937]],[logseq____"^15logseq____",[425,logseq____"^Vlogseq____",404,536870937]],[logseq____"^15logseq____",[425,logseq____"^Ulogseq____",39,536870937]],[logseq____"^15logseq____",[425,logseq____"^Ulogseq____",118,536870937]],[logseq____"^15logseq____",[425,logseq____"^Ulogseq____",122,536870937]],[logseq____"^15logseq____",[425,logseq____"^?logseq____",[logseq____"^ 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*(-1)\\\\\\\\\\nxlogseq____&logseq____>11\\n\\\\end{align}\\n\\\\begin{align}\\nL_3=\\\\left\\\\{x\\\\in\\\\Reals\\\\Big|xlogseq____>3,xlogseq____>11\\\\right\\\\}=(11,\\\\infty)\\n\\\\end{align}logseq____",536870937]],[logseq____"^15logseq____",[427,logseq____"^Ologseq____",logseq____"^16logseq____",536870937]],[logseq____"^15logseq____",[427,logseq____"^Flogseq____",398,536870937]],[logseq____"^15logseq____",[427,logseq____"^Xlogseq____",118,536870937]],[logseq____"^15logseq____",[427,logseq____"^Vlogseq____",395,536870937]],[logseq____"^15logseq____",[427,logseq____"^Ulogseq____",39,536870937]],[logseq____"^15logseq____",[427,logseq____"^Ulogseq____",112,536870937]],[logseq____"^15logseq____",[427,logseq____"^Ulogseq____",117,536870937]],[logseq____"^15logseq____",[427,logseq____"^Ulogseq____",118,536870937]],[logseq____"^15logseq____",[427,logseq____"^Ulogseq____",119,536870937]],[logseq____"^15logseq____",[427,logseq____"^Ulogseq____",122,536870937]],[logseq____"^15logseq____",[427,logseq____"^17logseq____",true,536870937]],[logseq____"^15logseq____",[427,logseq____"^;logseq____",logseq____"~u652fddf4-08c3-47fb-bb99-2d3396ecb00clogseq____",536870937]],[logseq____"^15logseq____",[428,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nM=\\\\left\\\\{\\\\underbrace{(x,y)}_\\\\text{Punkte 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2}\\\\quad 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aller rellen Zahlen zwischen 4 und $\\\\infty$ ohne 4 und 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}$[[Wahrheitswerteverlauf]]$\\\\text{ von }(p\\\\rightarrow q)\\\\land(q\\\\rightarrow p)$\\n|$p$|$q$||$p\\\\rightarrow q$|$q\\\\rightarrow p$|$(p\\\\rightarrow q)\\\\land(q\\\\rightarrow p)|$p\\\\leftrightarrow q$|\\n|-|-|-|-|-|-|\\n|f|f||w|w|w|w|\\n|f|w||w|f|f|f|\\n|w|f||f|w|f|f|\\n|w|w||w|w|w|w|logseq____",536870939]],[logseq____"^15logseq____",[440,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[440,logseq____"^Flogseq____",458,536870939]],[logseq____"^15logseq____",[440,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[440,logseq____"^Vlogseq____",446,536870939]],[logseq____"^15logseq____",[440,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[440,logseq____"^Ulogseq____",172,536870939]],[logseq____"^15logseq____",[440,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[440,logseq____"^Ulogseq____",186,536870939]],[logseq____"^15logseq____",[440,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[440,logseq____"^Hlogseq____",172,536870939]],[logseq____"^15logseq____",[440,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[440,logseq____"^;logseq____",logseq____"~u652fddf4-61f9-4a7c-bd18-eca86876031alogseq____",536870939]],[logseq____"^15logseq____",[441,logseq____"^Qlogseq____",logseq____"Weitere [[Beispiele]]logseq____",536870939]],[logseq____"^15logseq____",[441,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[441,logseq____"^Flogseq____",449,536870939]],[logseq____"^15logseq____",[441,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[441,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[441,logseq____"^Ulogseq____",39,536870939]],[logseq____"^15logseq____",[441,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[441,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[441,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[441,logseq____"^Hlogseq____",39,536870939]],[logseq____"^15logseq____",[441,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[441,logseq____"^;logseq____",logseq____"~u652fddf4-4b0f-455e-9e56-f19899c70052logseq____",536870939]],[logseq____"^15logseq____",[442,logseq____"^Qlogseq____",logseq____"### [[Tantologie]]logseq____",536870939]],[logseq____"^15logseq____",[442,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[442,logseq____"^Flogseq____",446,536870939]],[logseq____"^15logseq____",[442,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[442,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[442,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[442,logseq____"^Ulogseq____",178,536870939]],[logseq____"^15logseq____",[442,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[442,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[442,logseq____"^?logseq____",[logseq____"^ 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[[Aussagen]]logseq____",536870939]],[logseq____"^15logseq____",[445,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[445,logseq____"^Flogseq____",438,536870939]],[logseq____"^15logseq____",[445,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[445,logseq____"^Vlogseq____",169,536870939]],[logseq____"^15logseq____",[445,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[445,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[445,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870939]],[logseq____"^15logseq____",[445,logseq____"^Jlogseq____",[],536870939]],[logseq____"^15logseq____",[445,logseq____"^Hlogseq____",184,536870939]],[logseq____"^15logseq____",[445,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[445,logseq____"^;logseq____",logseq____"~u652fddf4-430c-46b1-859b-fc572136041flogseq____",536870939]],[logseq____"^15logseq____",[446,logseq____"^Qlogseq____",logseq____"### [[Wahrheitswerteverlauf]] einer [[Aussageformel]]logseq____",536870939]],[logseq____"^15logseq____",[446,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[446,logseq____"^Flogseq____",460,536870939]],[logseq____"^15logseq____",[446,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[446,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[446,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[446,logseq____"^Ulogseq____",172,536870939]],[logseq____"^15logseq____",[446,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[446,logseq____"^Ulogseq____",186,536870939]],[logseq____"^15logseq____",[446,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[446,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870939]],[logseq____"^15logseq____",[446,logseq____"^Jlogseq____",[],536870939]],[logseq____"^15logseq____",[446,logseq____"^Hlogseq____",172,536870939]],[logseq____"^15logseq____",[446,logseq____"^Hlogseq____",186,536870939]],[logseq____"^15logseq____",[446,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[446,logseq____"^;logseq____",logseq____"~u652fddf4-6a3f-49a7-ad7a-3853b2f7da35logseq____",536870939]],[logseq____"^15logseq____",[447,logseq____"^Qlogseq____",logseq____"Formel, die konstant $f$ istlogseq____",536870939]],[logseq____"^15logseq____",[447,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[447,logseq____"^Flogseq____",444,536870939]],[logseq____"^15logseq____",[447,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[447,logseq____"^Vlogseq____",444,536870939]],[logseq____"^15logseq____",[447,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[447,logseq____"^Ulogseq____",179,536870939]],[logseq____"^15logseq____",[447,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[447,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[447,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[447,logseq____"^;logseq____",logseq____"~u652fddf4-51b5-402e-9a75-02a19ed4b285logseq____",536870939]],[logseq____"^15logseq____",[448,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870939]],[logseq____"^15logseq____",[448,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[448,logseq____"^Flogseq____",451,536870939]],[logseq____"^15logseq____",[448,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[448,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[448,logseq____"^Ulogseq____",39,536870939]],[logseq____"^15logseq____",[448,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[448,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[448,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[448,logseq____"^Hlogseq____",39,536870939]],[logseq____"^15logseq____",[448,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[448,logseq____"^;logseq____",logseq____"~u652fddf4-71cb-4721-9eed-b69ec562d7c0logseq____",536870939]],[logseq____"^15logseq____",[449,logseq____"^Qlogseq____",logseq____"Alternativ: $p\\\\equiv q$, falls $p\\\\leftrightarrow q$ [[Tantologie]]logseq____",536870939]],[logseq____"^15logseq____",[449,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[449,logseq____"^Flogseq____",465,536870939]],[logseq____"^15logseq____",[449,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[449,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[449,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[449,logseq____"^Ulogseq____",178,536870939]],[logseq____"^15logseq____",[449,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[449,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[449,logseq____"^Hlogseq____",178,536870939]],[logseq____"^15logseq____",[449,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[449,logseq____"^;logseq____",logseq____"~u652fddf4-c6e3-4f5d-9c0e-470a697a50d8logseq____",536870939]],[logseq____"^15logseq____",[450,logseq____"^Qlogseq____",logseq____"## [[Verknüpfung von Aussagen]]logseq____",536870939]],[logseq____"^15logseq____",[450,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[450,logseq____"^Flogseq____",462,536870939]],[logseq____"^15logseq____",[450,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[450,logseq____"^Vlogseq____",445,536870939]],[logseq____"^15logseq____",[450,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[450,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[450,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[450,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870939]],[logseq____"^15logseq____",[450,logseq____"^Jlogseq____",[],536870939]],[logseq____"^15logseq____",[450,logseq____"^Hlogseq____",188,536870939]],[logseq____"^15logseq____",[450,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[450,logseq____"^;logseq____",logseq____"~u652fddf4-1776-4cb5-b5ac-a5b69ec8d3f8logseq____",536870939]],[logseq____"^15logseq____",[451,logseq____"^Qlogseq____",logseq____"Sind $p$ und $q$ [[Aussagen]], so auch\\n\\\\begin{align}\\n(p\\\\land q) logseq____&\\\\quadlogseq____& (\\logseq____"\\\\text{und}\\logseq____")\\\\\\\\\\n(p\\\\lor q) logseq____&logseq____& (\\logseq____"\\\\text{oder}\\logseq____")\\\\\\\\\\n(\\\\neg p) logseq____&logseq____& (\\logseq____"\\\\text{nicht}\\logseq____")\\\\\\\\\\n(p\\\\rightarrow q) logseq____&logseq____& (\\logseq____"\\\\text{impliziert}\\logseq____")\\\\\\\\\\n(p\\\\leftrightarrow q) logseq____&logseq____& (\\logseq____"\\\\text{genau dann wenn}\\logseq____")\\n\\\\end{align}logseq____",536870939]],[logseq____"^15logseq____",[451,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[451,logseq____"^Flogseq____",450,536870939]],[logseq____"^15logseq____",[451,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[451,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[451,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[451,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[451,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[451,logseq____"^Hlogseq____",184,536870939]],[logseq____"^15logseq____",[451,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[451,logseq____"^;logseq____",logseq____"~u652fddf4-8151-4085-aedd-9865894318e1logseq____",536870939]],[logseq____"^15logseq____",[452,logseq____"^Qlogseq____",logseq____"### [[Aussagenlogische Variable]]logseq____",536870939]],[logseq____"^15logseq____",[452,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[452,logseq____"^Flogseq____",461,536870939]],[logseq____"^15logseq____",[452,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[452,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[452,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[452,logseq____"^Ulogseq____",173,536870939]],[logseq____"^15logseq____",[452,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[452,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[452,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870939]],[logseq____"^15logseq____",[452,logseq____"^Jlogseq____",[],536870939]],[logseq____"^15logseq____",[452,logseq____"^Hlogseq____",173,536870939]],[logseq____"^15logseq____",[452,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[452,logseq____"^;logseq____",logseq____"~u652fddf4-f262-43ad-a4f8-857b67951e66logseq____",536870939]],[logseq____"^15logseq____",[453,logseq____"^Qlogseq____",logseq____"[Variable]([[Variablen]]), die den Wert $w$ oder $f$ annimmtlogseq____",536870939]],[logseq____"^15logseq____",[453,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[453,logseq____"^Flogseq____",452,536870939]],[logseq____"^15logseq____",[453,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[453,logseq____"^Vlogseq____",452,536870939]],[logseq____"^15logseq____",[453,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[453,logseq____"^Ulogseq____",173,536870939]],[logseq____"^15logseq____",[453,logseq____"^Ulogseq____",180,536870939]],[logseq____"^15logseq____",[453,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[453,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[453,logseq____"^Hlogseq____",180,536870939]],[logseq____"^15logseq____",[453,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[453,logseq____"^;logseq____",logseq____"~u652fddf4-1352-40a3-9a9b-451846273363logseq____",536870939]],[logseq____"^15logseq____",[454,logseq____"^Qlogseq____",logseq____"[[Wikipedia]]logseq____",536870939]],[logseq____"^15logseq____",[454,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[454,logseq____"^Flogseq____",457,536870939]],[logseq____"^15logseq____",[454,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[454,logseq____"^Vlogseq____",438,536870939]],[logseq____"^15logseq____",[454,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[454,logseq____"^Ulogseq____",174,536870939]],[logseq____"^15logseq____",[454,logseq____"^Hlogseq____",174,536870939]],[logseq____"^15logseq____",[454,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[454,logseq____"^;logseq____",logseq____"~u652fddf4-a3e1-4e34-a119-0b69cc7af410logseq____",536870939]],[logseq____"^15logseq____",[455,logseq____"^Qlogseq____",logseq____"### [[Aussageformel]]logseq____",536870939]],[logseq____"^15logseq____",[455,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[455,logseq____"^Flogseq____",452,536870939]],[logseq____"^15logseq____",[455,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[455,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[455,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[455,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[455,logseq____"^Ulogseq____",186,536870939]],[logseq____"^15logseq____",[455,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[455,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870939]],[logseq____"^15logseq____",[455,logseq____"^Jlogseq____",[],536870939]],[logseq____"^15logseq____",[455,logseq____"^Hlogseq____",186,536870939]],[logseq____"^15logseq____",[455,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[455,logseq____"^;logseq____",logseq____"~u652fddf4-e5b1-4ebe-ac36-fd3f0ee2b313logseq____",536870939]],[logseq____"^15logseq____",[456,logseq____"^Qlogseq____",logseq____"Zuordnung von $w/f$ an jede [Variable]([[Variablen]]) einer [[Aussageformel]]logseq____",536870939]],[logseq____"^15logseq____",[456,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[456,logseq____"^Flogseq____",460,536870939]],[logseq____"^15logseq____",[456,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[456,logseq____"^Vlogseq____",460,536870939]],[logseq____"^15logseq____",[456,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[456,logseq____"^Ulogseq____",180,536870939]],[logseq____"^15logseq____",[456,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[456,logseq____"^Ulogseq____",186,536870939]],[logseq____"^15logseq____",[456,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[456,logseq____"^Ulogseq____",190,536870939]],[logseq____"^15logseq____",[456,logseq____"^Hlogseq____",180,536870939]],[logseq____"^15logseq____",[456,logseq____"^Hlogseq____",186,536870939]],[logseq____"^15logseq____",[456,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[456,logseq____"^;logseq____",logseq____"~u652fddf4-5b58-4165-b03f-88f0d039730clogseq____",536870939]],[logseq____"^15logseq____",[457,logseq____"^Qlogseq____",logseq____"C. Meinel, M. Mundhenk, [[\\logseq____"Mathematische Grundlagen der Informatik\\logseq____", 5. Auflage 2011]]logseq____",536870939]],[logseq____"^15logseq____",[457,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[457,logseq____"^Flogseq____",438,536870939]],[logseq____"^15logseq____",[457,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[457,logseq____"^Vlogseq____",438,536870939]],[logseq____"^15logseq____",[457,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[457,logseq____"^Ulogseq____",177,536870939]],[logseq____"^15logseq____",[457,logseq____"^Hlogseq____",177,536870939]],[logseq____"^15logseq____",[457,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[457,logseq____"^;logseq____",logseq____"~u652fddf4-4f37-48fd-863b-9d851e906248logseq____",536870939]],[logseq____"^15logseq____",[458,logseq____"^Qlogseq____",logseq____"Jede mögliche [[Belegung]] wird ein [[Wahrheitswert]] der [[Formel]] zugeordnet (nach [[Tabellenregeln]])logseq____",536870939]],[logseq____"^15logseq____",[458,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[458,logseq____"^Flogseq____",446,536870939]],[logseq____"^15logseq____",[458,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[458,logseq____"^Vlogseq____",446,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",172,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",175,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",176,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",183,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",186,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[458,logseq____"^Ulogseq____",190,536870939]],[logseq____"^15logseq____",[458,logseq____"^Hlogseq____",175,536870939]],[logseq____"^15logseq____",[458,logseq____"^Hlogseq____",176,536870939]],[logseq____"^15logseq____",[458,logseq____"^Hlogseq____",183,536870939]],[logseq____"^15logseq____",[458,logseq____"^Hlogseq____",190,536870939]],[logseq____"^15logseq____",[458,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[458,logseq____"^;logseq____",logseq____"~u652fddf4-10f1-4e6d-be19-ecad22a41ce2logseq____",536870939]],[logseq____"^15logseq____",[459,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\np\\\\land q\\\\lor r\\\\\\\\\\n(p\\\\land q)\\\\lor r\\\\\\\\\\np\\\\land (q\\\\lor r)\\\\\\\\\\n\\\\end{align}logseq____",536870939]],[logseq____"^15logseq____",[459,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[459,logseq____"^Flogseq____",448,536870939]],[logseq____"^15logseq____",[459,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[459,logseq____"^Vlogseq____",448,536870939]],[logseq____"^15logseq____",[459,logseq____"^Ulogseq____",39,536870939]],[logseq____"^15logseq____",[459,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[459,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[459,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[459,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[459,logseq____"^;logseq____",logseq____"~u652fddf4-dddd-4681-bf79-fb68953d55fdlogseq____",536870939]],[logseq____"^15logseq____",[460,logseq____"^Qlogseq____",logseq____"### [[Belegung]] (der [[Variablen]])logseq____",536870939]],[logseq____"^15logseq____",[460,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[460,logseq____"^Flogseq____",455,536870939]],[logseq____"^15logseq____",[460,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[460,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[460,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[460,logseq____"^Ulogseq____",180,536870939]],[logseq____"^15logseq____",[460,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[460,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[460,logseq____"^Ulogseq____",190,536870939]],[logseq____"^15logseq____",[460,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870939]],[logseq____"^15logseq____",[460,logseq____"^Jlogseq____",[],536870939]],[logseq____"^15logseq____",[460,logseq____"^Hlogseq____",180,536870939]],[logseq____"^15logseq____",[460,logseq____"^Hlogseq____",190,536870939]],[logseq____"^15logseq____",[460,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[460,logseq____"^;logseq____",logseq____"~u652fddf4-6144-494b-91c3-87800fd85a5elogseq____",536870939]],[logseq____"^15logseq____",[461,logseq____"^Qlogseq____",logseq____"Soweit die \\logseq____"[[Syntax]]\\logseq____", jetzt die [[Semantik]]. Die [Wahrheitswerte]([[Wahrheitswert]])\\n ergeben sich aus den Wahrheitswerten von $p$,$q$ gemäß folgender Tabelle ([[Tabellenregeln]]).\\n|$p$|$q$||$p\\\\land q$|$p\\\\lor q$|$\\\\neg p$|$p\\\\rightarrow q$|$p\\\\leftrightarrow q$|\\n|---|----||-----------|---------|---------|-----------------|---------------------|\\n|f|f||f|f|w|w|w|\\n|f|w||f|w|w|w|f|\\n|w|f||f|w|f|f|f|\\n|w|w||w|w|f|w|w|logseq____",536870939]],[logseq____"^15logseq____",[461,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[461,logseq____"^Flogseq____",448,536870939]],[logseq____"^15logseq____",[461,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[461,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[461,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[461,logseq____"^Ulogseq____",176,536870939]],[logseq____"^15logseq____",[461,logseq____"^Ulogseq____",181,536870939]],[logseq____"^15logseq____",[461,logseq____"^Ulogseq____",183,536870939]],[logseq____"^15logseq____",[461,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[461,logseq____"^Ulogseq____",187,536870939]],[logseq____"^15logseq____",[461,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[461,logseq____"^Hlogseq____",176,536870939]],[logseq____"^15logseq____",[461,logseq____"^Hlogseq____",181,536870939]],[logseq____"^15logseq____",[461,logseq____"^Hlogseq____",183,536870939]],[logseq____"^15logseq____",[461,logseq____"^Hlogseq____",187,536870939]],[logseq____"^15logseq____",[461,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[461,logseq____"^;logseq____",logseq____"~u652fddf4-cc6e-475a-a244-d80315c0f0f1logseq____",536870939]],[logseq____"^15logseq____",[462,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiele}$\\n\\\\begin{align}\\nlogseq____&\\logseq____"5\\\\text{ ist prim}\\logseq____" logseq____&\\\\quadlogseq____& (w)\\\\\\\\\\n\\\\notag\\\\\\\\\\nlogseq____&\\logseq____"4\\\\text{ ist prim}\\logseq____" logseq____&logseq____& (f)\\\\\\\\\\n\\\\notag\\\\\\\\\\nlogseq____&\\logseq____"\\\\text{Jede gerade Zahl }\\\\ge4\\\\notag\\\\\\\\\\nlogseq____&\\\\text{ ist Summe zweier Primzahlen}\\logseq____" logseq____&logseq____& (\\\\text{Aussage, }w\\\\text{ oder }f)\\\\notag\\\\\\\\\\nlogseq____&(\\\\text{Vermutung von Goldback 1742})\\\\text{, }logseq____&logseq____&\\\\text{richtig für gerade Zahlen bis }4*10^{18}\\\\\\\\\\n\\\\notag\\\\\\\\\\nlogseq____&\\logseq____"\\\\text{Dieser Satz ist falsch}\\logseq____" logseq____&logseq____& (\\\\text{keine Aussage})\\\\\\\\\\n\\\\notag\\\\\\\\\\nlogseq____&\\logseq____"\\\\text{Die Gleichung } x^2+y^2=z^2 \\\\notag\\\\\\\\\\nlogseq____&\\\\text{ hat eine Lösung }x,y,z\\\\notag\\\\\\\\\\nlogseq____&\\\\text{ aus positiven ganzen Zahlen}\\logseq____" logseq____&logseq____& (w\\\\text{, z.B. }x=3,y=4,z=5)\\\\\\\\\\n\\\\notag\\\\\\\\\\nlogseq____&\\logseq____"\\\\text{Für }n\\\\ge3\\\\text{ hat die Gleichung }\\\\notag\\\\\\\\\\nlogseq____&x^n+y^n=z^n\\\\text{ eine Lösung}logseq____&logseq____& (f\\\\text{, wie von Fermat 1640 vermutet}\\\\notag\\\\\\\\\\nlogseq____&x,y,z \\\\text{aus positiven ganzen Zahlen}\\logseq____" logseq____&logseq____&\\\\text{und von Wiles 1994 bewiesen})\\\\\\\\\\n\\\\notag\\\\\\\\\\nlogseq____&\\logseq____"\\\\text{Gott ist tot}\\logseq____" logseq____&logseq____& (\\\\text{Aussage? Wohl kaum?})\\\\\\\\\\n\\\\notag\\\\\\\\\\nlogseq____&\\logseq____"\\\\text{Nietzsche ist tot}\\logseq____" logseq____&logseq____& (\\\\text{Aussage?}) \\\\\\\\\\n\\\\notag\\\\\\\\\\nlogseq____&\\logseq____"\\\\text{Die Länder einer Landkarte lassen sich}\\\\notag\\\\\\\\\\nlogseq____&\\\\text{so mit nur vier Farben färben,}\\\\notag\\\\\\\\\\nlogseq____&\\\\text{dass Länder mit einer gemeinsamen}\\\\notag\\\\\\\\\\nlogseq____&\\\\text{Grenzlienie verschieden gefärbt sind.}\\logseq____" logseq____&logseq____& (\\\\text{Aussagge; }w\\\\text{, sog 4-Farben-Satz})\\n\\\\end{align}logseq____",536870939]],[logseq____"^15logseq____",[462,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[462,logseq____"^Flogseq____",443,536870939]],[logseq____"^15logseq____",[462,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[462,logseq____"^Vlogseq____",445,536870939]],[logseq____"^15logseq____",[462,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[462,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[462,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[462,logseq____"^;logseq____",logseq____"~u652fddf4-d1ad-4f8e-a062-c43d7f73ad06logseq____",536870939]],[logseq____"^15logseq____",[463,logseq____"^Qlogseq____",logseq____"Entsteht durch sukzessive [[Verknüpfungen]] wie oben an Variablen ergibt #FIXMElogseq____",536870939]],[logseq____"^15logseq____",[463,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[463,logseq____"^Flogseq____",455,536870939]],[logseq____"^15logseq____",[463,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[463,logseq____"^Vlogseq____",455,536870939]],[logseq____"^15logseq____",[463,logseq____"^Ulogseq____",46,536870939]],[logseq____"^15logseq____",[463,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[463,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[463,logseq____"^Ulogseq____",185,536870939]],[logseq____"^15logseq____",[463,logseq____"^Ulogseq____",186,536870939]],[logseq____"^15logseq____",[463,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[463,logseq____"^Hlogseq____",46,536870939]],[logseq____"^15logseq____",[463,logseq____"^Hlogseq____",185,536870939]],[logseq____"^15logseq____",[463,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[463,logseq____"^;logseq____",logseq____"~u652fddf4-4471-4607-badc-dd926e52133dlogseq____",536870939]],[logseq____"^15logseq____",[464,logseq____"^Qlogseq____",logseq____"Formel, die konstant $w$ istlogseq____",536870939]],[logseq____"^15logseq____",[464,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[464,logseq____"^Flogseq____",442,536870939]],[logseq____"^15logseq____",[464,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[464,logseq____"^Vlogseq____",442,536870939]],[logseq____"^15logseq____",[464,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[464,logseq____"^Ulogseq____",178,536870939]],[logseq____"^15logseq____",[464,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[464,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[464,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[464,logseq____"^;logseq____",logseq____"~u652fddf4-5669-4318-bab1-02f88e8fd1a7logseq____",536870939]],[logseq____"^15logseq____",[465,logseq____"^Qlogseq____",logseq____"Formeln $p,q$ [logisch äquivalent]([[Logisch Äquivalent]]), wenn sie den gleichen [[Wahrheitswerteverlauf]] haben.\\n[[Schreibweise]]: $p\\\\equiv q$logseq____",536870939]],[logseq____"^15logseq____",[465,logseq____"^Ologseq____",logseq____"^16logseq____",536870939]],[logseq____"^15logseq____",[465,logseq____"^Flogseq____",444,536870939]],[logseq____"^15logseq____",[465,logseq____"^Xlogseq____",169,536870939]],[logseq____"^15logseq____",[465,logseq____"^Vlogseq____",450,536870939]],[logseq____"^15logseq____",[465,logseq____"^Ulogseq____",169,536870939]],[logseq____"^15logseq____",[465,logseq____"^Ulogseq____",172,536870939]],[logseq____"^15logseq____",[465,logseq____"^Ulogseq____",182,536870939]],[logseq____"^15logseq____",[465,logseq____"^Ulogseq____",184,536870939]],[logseq____"^15logseq____",[465,logseq____"^Ulogseq____",188,536870939]],[logseq____"^15logseq____",[465,logseq____"^Ulogseq____",189,536870939]],[logseq____"^15logseq____",[465,logseq____"^Hlogseq____",172,536870939]],[logseq____"^15logseq____",[465,logseq____"^Hlogseq____",182,536870939]],[logseq____"^15logseq____",[465,logseq____"^Hlogseq____",189,536870939]],[logseq____"^15logseq____",[465,logseq____"^17logseq____",true,536870939]],[logseq____"^15logseq____",[465,logseq____"^;logseq____",logseq____"~u652fddf4-bac1-4cb1-8ce8-e8b78f944ca0logseq____",536870939]],[logseq____"^15logseq____",[467,logseq____"^Qlogseq____",logseq____"[[Satz]] mit [[Variablen]] $x_1,\\\\cdots,x_n$, der bei Ersetzung jedes $x_j$ durch ein [Objekt]([[Objekt (Logik)]]) aus $U_j$ stets zu einer [Aussage]([[Aussagen]]) wirdlogseq____",536870940]],[logseq____"^15logseq____",[467,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[467,logseq____"^Flogseq____",494,536870940]],[logseq____"^15logseq____",[467,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[467,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[467,logseq____"^Ulogseq____",180,536870940]],[logseq____"^15logseq____",[467,logseq____"^Ulogseq____",184,536870940]],[logseq____"^15logseq____",[467,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[467,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[467,logseq____"^Ulogseq____",241,536870940]],[logseq____"^15logseq____",[467,logseq____"^Ulogseq____",244,536870940]],[logseq____"^15logseq____",[467,logseq____"^Hlogseq____",180,536870940]],[logseq____"^15logseq____",[467,logseq____"^Hlogseq____",184,536870940]],[logseq____"^15logseq____",[467,logseq____"^Hlogseq____",241,536870940]],[logseq____"^15logseq____",[467,logseq____"^Hlogseq____",244,536870940]],[logseq____"^15logseq____",[467,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[467,logseq____"^;logseq____",logseq____"~u652fddf5-05e2-416e-9fc5-8bd17b77f89dlogseq____",536870940]],[logseq____"^15logseq____",[468,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\logseq____"\\\\text{x ist prim}\\logseq____" \\\\quad logseq____& \\\\text{ist AF mit Variablen x über den natürlichen Zahlen}\\\\\\\\\\n\\logseq____"\\\\underbrace{x}_{x_1}logseq____<\\\\underbrace{y}_{x_2}\\logseq____"\\\\quad logseq____&\\\\text{ist AF mit Variablen }x,y\\\\text{ über die reellen Zahlen } \\\\underbrace{\\\\Reals}_{U_1},\\\\underbrace{\\\\Reals}_{U_2}\\\\\\\\\\n\\logseq____"\\\\text{x ist wahr}\\logseq____"\\\\quadlogseq____&\\\\text{ist AF mit Variablen x über dem Universum der Aussagen}\\n\\\\end{align}logseq____",536870940]],[logseq____"^15logseq____",[468,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[468,logseq____"^Flogseq____",504,536870940]],[logseq____"^15logseq____",[468,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[468,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[468,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[468,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[468,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[468,logseq____"^;logseq____",logseq____"~u652fddf5-4cb3-4fea-8657-a8643b797c64logseq____",536870940]],[logseq____"^15logseq____",[469,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist falsch}\\n\\\\end{align}\\n[Beweis]([[Beweise]]):logseq____",536870940]],[logseq____"^15logseq____",[469,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[469,logseq____"^Flogseq____",492,536870940]],[logseq____"^15logseq____",[469,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[469,logseq____"^Vlogseq____",475,536870940]],[logseq____"^15logseq____",[469,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[469,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[469,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[469,logseq____"^Ulogseq____",231,536870940]],[logseq____"^15logseq____",[469,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[469,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[469,logseq____"^Hlogseq____",231,536870940]],[logseq____"^15logseq____",[469,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[469,logseq____"^;logseq____",logseq____"~u652fddf5-bd7f-493a-860e-8397a8d6f8c9logseq____",536870940]],[logseq____"^15logseq____",[470,logseq____"^Qlogseq____",logseq____"Wäre $y$ aus $\\\\Reals$ mit $xlogseq____<y$ für alle $x$ aus $\\\\Reals$, so insbesondere auch $ylogseq____<y$, Widerspruch. $\\\\blacksquare$logseq____",536870940]],[logseq____"^15logseq____",[470,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[470,logseq____"^Flogseq____",469,536870940]],[logseq____"^15logseq____",[470,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[470,logseq____"^Vlogseq____",469,536870940]],[logseq____"^15logseq____",[470,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[470,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[470,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[470,logseq____"^Ulogseq____",231,536870940]],[logseq____"^15logseq____",[470,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[470,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[470,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[470,logseq____"^;logseq____",logseq____"~u652fddf5-137b-40ea-9a7c-02472802becclogseq____",536870940]],[logseq____"^15logseq____",[471,logseq____"^Qlogseq____",logseq____"Alternativschreibweisen:\\n$\\\\forall x: p(x)$ auch $\\\\bigwedge x:p(x)$\\n$\\\\exists x:p(x)$ auch $\\\\bigvee x:p(x)$logseq____",536870940]],[logseq____"^15logseq____",[471,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[471,logseq____"^Flogseq____",498,536870940]],[logseq____"^15logseq____",[471,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[471,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[471,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[471,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[471,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[471,logseq____"^;logseq____",logseq____"~u652fddf5-d680-4ff1-bae5-d83a52345beelogseq____",536870940]],[logseq____"^15logseq____",[472,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nlogseq____&\\\\forall x: \\\\forall y: p(x, y) \\\\equiv \\\\forall y: \\\\forall x: p(x,y)\\\\\\\\\\nlogseq____&\\\\exists x:\\\\exists y:p(x,y)\\\\equiv\\\\exists y:\\\\exists x:p(x,y)\\\\\\\\\\nlogseq____&[\\\\forall x_1:\\\\forall x_2: \\\\cdots:\\\\forall x_n:p(x_1,x_2,\\\\cdots,x_n)\\\\notag\\\\\\\\\\nlogseq____&\\\\equiv\\\\forall_{x_{\\\\pi(1)}}:\\\\forall_{x_{\\\\pi(1)}}:\\\\cdots:\\\\forall_{x_{\\\\pi(1)}}:p(x_1,x_2,\\\\cdots,x_n)\\n\\\\end{align}logseq____",536870940]],[logseq____"^15logseq____",[472,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[472,logseq____"^Flogseq____",474,536870940]],[logseq____"^15logseq____",[472,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[472,logseq____"^Vlogseq____",474,536870940]],[logseq____"^15logseq____",[472,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[472,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[472,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[472,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[472,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[472,logseq____"^;logseq____",logseq____"~u652fddf5-83d8-4b84-8535-50d2ae69d6fclogseq____",536870940]],[logseq____"^15logseq____",[473,logseq____"^Qlogseq____",logseq____"Sei $p(x,y)$ AF in Variablen $x,y$ über $U_1,U_2$\\n\\\\begin{align}\\n\\\\underbrace{\\\\forall x: \\\\underbrace{\\\\exists y: p(x,y)}_{\\\\text{AF in den Variablen }x\\\\text{ über }U_1}}_{\\\\text{Aussage}}\\n\\\\end{align}logseq____",536870940]],[logseq____"^15logseq____",[473,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[473,logseq____"^Flogseq____",497,536870940]],[logseq____"^15logseq____",[473,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[473,logseq____"^Vlogseq____",497,536870940]],[logseq____"^15logseq____",[473,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[473,logseq____"^Ulogseq____",220,536870940]],[logseq____"^15logseq____",[473,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[473,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[473,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[473,logseq____"^;logseq____",logseq____"~u652fddf5-268b-49f4-beb8-be94d2fd890dlogseq____",536870940]],[logseq____"^15logseq____",[474,logseq____"^Qlogseq____",logseq____"[Gleichartige]([[Gleichartig]]) [[Quantoren]] dürfen vertauscht werdenlogseq____",536870940]],[logseq____"^15logseq____",[474,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[474,logseq____"^Flogseq____",495,536870940]],[logseq____"^15logseq____",[474,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[474,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[474,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[474,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[474,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[474,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[474,logseq____"^Hlogseq____",228,536870940]],[logseq____"^15logseq____",[474,logseq____"^Hlogseq____",232,536870940]],[logseq____"^15logseq____",[474,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[474,logseq____"^;logseq____",logseq____"~u652fddf5-4763-41b9-a763-26ac51944e91logseq____",536870940]],[logseq____"^15logseq____",[475,logseq____"^Qlogseq____",logseq____"Betrachte $\\logseq____"xlogseq____<y\\logseq____"$ als AF über $\\\\Reals, \\\\Reals$logseq____",536870940]],[logseq____"^15logseq____",[475,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[475,logseq____"^Flogseq____",486,536870940]],[logseq____"^15logseq____",[475,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[475,logseq____"^Vlogseq____",486,536870940]],[logseq____"^15logseq____",[475,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[475,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[475,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[475,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[475,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[475,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[475,logseq____"^;logseq____",logseq____"~u652fddf5-4a99-4d3a-8390-094f5fb8900dlogseq____",536870940]],[logseq____"^15logseq____",[476,logseq____"^Qlogseq____",logseq____"see: [[Große Azzoziativgesetze]] [[Große de Morgansche Regeln]] [[Negationsregeln]]logseq____",536870940]],[logseq____"^15logseq____",[476,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[476,logseq____"^Flogseq____",501,536870940]],[logseq____"^15logseq____",[476,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[476,logseq____"^Vlogseq____",501,536870940]],[logseq____"^15logseq____",[476,logseq____"^Ulogseq____",220,536870940]],[logseq____"^15logseq____",[476,logseq____"^Ulogseq____",223,536870940]],[logseq____"^15logseq____",[476,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[476,logseq____"^Ulogseq____",235,536870940]],[logseq____"^15logseq____",[476,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[476,logseq____"^Ulogseq____",240,536870940]],[logseq____"^15logseq____",[476,logseq____"^Hlogseq____",223,536870940]],[logseq____"^15logseq____",[476,logseq____"^Hlogseq____",235,536870940]],[logseq____"^15logseq____",[476,logseq____"^Hlogseq____",240,536870940]],[logseq____"^15logseq____",[476,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[476,logseq____"^;logseq____",logseq____"~u652fddf5-e75b-472a-ad93-de34a607a342logseq____",536870940]],[logseq____"^15logseq____",[477,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(p\\\\land q)\\\\land r logseq____&\\\\equiv p\\\\land(q\\\\land r),\\\\quadlogseq____&\\\\text{schreibe daher: } p\\\\land q\\\\land r,\\\\quad \\\\land \\\\text{ assoziativ}\\\\\\\\\\n(p \\\\lor q) \\\\lor r logseq____&\\\\equiv p \\\\lor (q\\\\lor r),\\\\quadlogseq____&\\\\text{schreibe daher: } p\\\\lor q\\\\lor r,\\\\quad \\\\lor \\\\text{ assoziativ}\\\\\\\\\\np \\\\land q logseq____&\\\\equiv q \\\\land p,\\\\quad \\\\land logseq____&\\\\text{ kommutativ}\\\\\\\\\\np \\\\lor q logseq____&\\\\equiv q \\\\lor p,\\\\quad \\\\lor logseq____&\\\\text{ kommutativ}\\\\\\\\\\n\\\\neg(\\\\neg p) logseq____&\\\\equiv p, \\\\quad \\\\neg logseq____&\\\\text{ idenpotent}\\\\\\\\\\np\\\\land(q\\\\lor r)logseq____&\\\\equiv (p\\\\land q)\\\\lor(p\\\\land r);\\\\notag\\\\\\\\\\np\\\\lor(q\\\\land r)logseq____&\\\\equiv (p\\\\lor q)\\\\lor(p\\\\land r),\\\\quad logseq____&\\\\land,\\\\lor\\\\text{ wechselseitig distributiv}\\\\\\\\\\n\\\\neg(p\\\\land q)logseq____&\\\\equiv(\\\\neg p)\\\\lor(\\\\neg q);\\\\notag\\\\\\\\\\n\\\\neg(p\\\\land q)logseq____&\\\\equiv(\\\\neg p)\\\\land (\\\\neg q), \\\\quad logseq____&\\\\text{ (kleine) de Morgansche Regeln}\\\\\\\\\\n\\\\end{align}logseq____",536870940]],[logseq____"^15logseq____",[477,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[477,logseq____"^Flogseq____",489,536870940]],[logseq____"^15logseq____",[477,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[477,logseq____"^Vlogseq____",227,536870940]],[logseq____"^15logseq____",[477,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[477,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[477,logseq____"^;logseq____",logseq____"~u652fddf5-f180-44bd-a647-08c6de301b07logseq____",536870940]],[logseq____"^15logseq____",[478,logseq____"^Qlogseq____",logseq____"# [[Aussageform]]logseq____",536870940]],[logseq____"^15logseq____",[478,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[478,logseq____"^Flogseq____",477,536870940]],[logseq____"^15logseq____",[478,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[478,logseq____"^Vlogseq____",227,536870940]],[logseq____"^15logseq____",[478,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[478,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[478,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870940]],[logseq____"^15logseq____",[478,logseq____"^Jlogseq____",[],536870940]],[logseq____"^15logseq____",[478,logseq____"^Hlogseq____",238,536870940]],[logseq____"^15logseq____",[478,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[478,logseq____"^;logseq____",logseq____"~u652fddf5-2bd0-48c9-8d49-14c342821176logseq____",536870940]],[logseq____"^15logseq____",[479,logseq____"^Qlogseq____",logseq____"Sei $x$ aus $\\\\Reals$ beliebig. Setze $y:=x+1$ (ist aus $\\\\Reals$). Dann ist $xlogseq____<x+1=y$ also $xlogseq____<y$. $\\\\blacksquare$logseq____",536870940]],[logseq____"^15logseq____",[479,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[479,logseq____"^Flogseq____",492,536870940]],[logseq____"^15logseq____",[479,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[479,logseq____"^Vlogseq____",492,536870940]],[logseq____"^15logseq____",[479,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[479,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[479,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[479,logseq____"^Ulogseq____",231,536870940]],[logseq____"^15logseq____",[479,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[479,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[479,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[479,logseq____"^;logseq____",logseq____"~u652fddf5-80e6-485e-88d3-fb30e66572b8logseq____",536870940]],[logseq____"^15logseq____",[480,logseq____"^Qlogseq____",logseq____"see: [[assoziativ]] [[kommutativ]] [[idenpotent]] [[wechselseitig distributiv]] [[distributiv]] [[de Morgansche Regeln]]logseq____",536870940]],[logseq____"^15logseq____",[480,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[480,logseq____"^Flogseq____",477,536870940]],[logseq____"^15logseq____",[480,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[480,logseq____"^Vlogseq____",477,536870940]],[logseq____"^15logseq____",[480,logseq____"^Ulogseq____",221,536870940]],[logseq____"^15logseq____",[480,logseq____"^Ulogseq____",226,536870940]],[logseq____"^15logseq____",[480,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[480,logseq____"^Ulogseq____",229,536870940]],[logseq____"^15logseq____",[480,logseq____"^Ulogseq____",230,536870940]],[logseq____"^15logseq____",[480,logseq____"^Ulogseq____",236,536870940]],[logseq____"^15logseq____",[480,logseq____"^Ulogseq____",237,536870940]],[logseq____"^15logseq____",[480,logseq____"^Hlogseq____",221,536870940]],[logseq____"^15logseq____",[480,logseq____"^Hlogseq____",226,536870940]],[logseq____"^15logseq____",[480,logseq____"^Hlogseq____",229,536870940]],[logseq____"^15logseq____",[480,logseq____"^Hlogseq____",230,536870940]],[logseq____"^15logseq____",[480,logseq____"^Hlogseq____",236,536870940]],[logseq____"^15logseq____",[480,logseq____"^Hlogseq____",237,536870940]],[logseq____"^15logseq____",[480,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[480,logseq____"^;logseq____",logseq____"~u652fddf5-155c-413d-9082-2b142a404472logseq____",536870940]],[logseq____"^15logseq____",[481,logseq____"^Qlogseq____",logseq____"Nach Ersetzung von $x$ in der AF $\\logseq____"xlogseq____<y\\logseq____"$ durch eine konkrete Zahl entsteht eine AF in der Variablen $y$, z. Bsp. $\\logseq____"17logseq____<y\\logseq____"$logseq____",536870940]],[logseq____"^15logseq____",[481,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[481,logseq____"^Flogseq____",468,536870940]],[logseq____"^15logseq____",[481,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[481,logseq____"^Vlogseq____",468,536870940]],[logseq____"^15logseq____",[481,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[481,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[481,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[481,logseq____"^;logseq____",logseq____"~u652fddf5-c95b-4009-af2e-2994c853a49alogseq____",536870940]],[logseq____"^15logseq____",[482,logseq____"^Qlogseq____",logseq____"$\\\\forall x:p(x)$ ist die [Aussage]([[Aussagen]]): $\\logseq____"\\\\text{Für alle x aus U ist p(x) wahr}\\logseq____"\\nIhr [[Wahrheitswert]] ist $\\\\begin{cases}w\\\\quadlogseq____&\\\\text{fallls p(x) w für alle x aus U}\\\\\\\\f\\\\quadlogseq____&\\\\text{falls p(x) f für (wenigstens) ein x aus U}\\\\end{cases}$logseq____",536870940]],[logseq____"^15logseq____",[482,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[482,logseq____"^Flogseq____",503,536870940]],[logseq____"^15logseq____",[482,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[482,logseq____"^Vlogseq____",503,536870940]],[logseq____"^15logseq____",[482,logseq____"^Ulogseq____",176,536870940]],[logseq____"^15logseq____",[482,logseq____"^Ulogseq____",184,536870940]],[logseq____"^15logseq____",[482,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[482,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[482,logseq____"^Hlogseq____",176,536870940]],[logseq____"^15logseq____",[482,logseq____"^Hlogseq____",184,536870940]],[logseq____"^15logseq____",[482,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[482,logseq____"^;logseq____",logseq____"~u652fddf5-46fe-464a-a01e-d075eb086c40logseq____",536870940]],[logseq____"^15logseq____",[483,logseq____"^Qlogseq____",logseq____"#FIXMElogseq____",536870940]],[logseq____"^15logseq____",[483,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[483,logseq____"^Flogseq____",471,536870940]],[logseq____"^15logseq____",[483,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[483,logseq____"^Vlogseq____",471,536870940]],[logseq____"^15logseq____",[483,logseq____"^Ulogseq____",46,536870940]],[logseq____"^15logseq____",[483,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[483,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[483,logseq____"^Hlogseq____",46,536870940]],[logseq____"^15logseq____",[483,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[483,logseq____"^;logseq____",logseq____"~u652fddf5-9df6-405a-808d-640ec6f719b2logseq____",536870940]],[logseq____"^15logseq____",[484,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870940]],[logseq____"^15logseq____",[484,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[484,logseq____"^Flogseq____",474,536870940]],[logseq____"^15logseq____",[484,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[484,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[484,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[484,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[484,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[484,logseq____"^Hlogseq____",39,536870940]],[logseq____"^15logseq____",[484,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[484,logseq____"^;logseq____",logseq____"~u652fddf5-9b34-41b7-9996-5a505df07a4dlogseq____",536870940]],[logseq____"^15logseq____",[485,logseq____"^Qlogseq____",logseq____"Die beiden Quantoren $\\\\forall, \\\\exists$ erweitern die [Sprache]([[Sprachen]]) der [[Aussagenlogik]]logseq____",536870940]],[logseq____"^15logseq____",[485,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[485,logseq____"^Flogseq____",503,536870940]],[logseq____"^15logseq____",[485,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[485,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[485,logseq____"^Ulogseq____",224,536870940]],[logseq____"^15logseq____",[485,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[485,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[485,logseq____"^Ulogseq____",243,536870940]],[logseq____"^15logseq____",[485,logseq____"^Hlogseq____",224,536870940]],[logseq____"^15logseq____",[485,logseq____"^Hlogseq____",243,536870940]],[logseq____"^15logseq____",[485,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[485,logseq____"^;logseq____",logseq____"~u652fddf5-74fb-465b-a06d-feffcb800367logseq____",536870940]],[logseq____"^15logseq____",[486,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870940]],[logseq____"^15logseq____",[486,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[486,logseq____"^Flogseq____",472,536870940]],[logseq____"^15logseq____",[486,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[486,logseq____"^Vlogseq____",474,536870940]],[logseq____"^15logseq____",[486,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[486,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[486,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[486,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[486,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[486,logseq____"^Hlogseq____",39,536870940]],[logseq____"^15logseq____",[486,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[486,logseq____"^;logseq____",logseq____"~u652fddf5-7b87-4c87-b597-13b1e7d30085logseq____",536870940]],[logseq____"^15logseq____",[487,logseq____"^Qlogseq____",logseq____"see: [[Allquantor]] [[Existenzquantor]]logseq____",536870940]],[logseq____"^15logseq____",[487,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[487,logseq____"^Flogseq____",488,536870940]],[logseq____"^15logseq____",[487,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[487,logseq____"^Vlogseq____",488,536870940]],[logseq____"^15logseq____",[487,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[487,logseq____"^Ulogseq____",233,536870940]],[logseq____"^15logseq____",[487,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[487,logseq____"^Ulogseq____",239,536870940]],[logseq____"^15logseq____",[487,logseq____"^Hlogseq____",233,536870940]],[logseq____"^15logseq____",[487,logseq____"^Hlogseq____",239,536870940]],[logseq____"^15logseq____",[487,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[487,logseq____"^;logseq____",logseq____"~u652fddf5-dd23-4d12-8f00-8b65aacf7b42logseq____",536870940]],[logseq____"^15logseq____",[488,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\underbrace{\\\\forall}_\\\\text{Allquantor} x: p(x) logseq____&\\\\equiv p(O_1) \\\\land p(O_2) \\\\land \\\\cdots \\\\land p(O_m)\\\\\\\\\\n\\\\underbrace{\\\\exists}_\\\\text{Existenzquantor} x: p(x) logseq____&\\\\equiv p(0_1) \\\\lor p(O_2) \\\\lor \\\\cdots \\\\lor p(0_m)\\\\\\\\\\n\\\\end{align}logseq____",536870940]],[logseq____"^15logseq____",[488,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[488,logseq____"^Flogseq____",500,536870940]],[logseq____"^15logseq____",[488,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[488,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[488,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[488,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[488,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[488,logseq____"^;logseq____",logseq____"~u652fddf5-61b4-4659-aeb7-572ef89e88aflogseq____",536870940]],[logseq____"^15logseq____",[489,logseq____"^Qlogseq____",logseq____"# [[Regeln]]logseq____",536870940]],[logseq____"^15logseq____",[489,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[489,logseq____"^Flogseq____",227,536870940]],[logseq____"^15logseq____",[489,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[489,logseq____"^Vlogseq____",227,536870940]],[logseq____"^15logseq____",[489,logseq____"^Ulogseq____",220,536870940]],[logseq____"^15logseq____",[489,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[489,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870940]],[logseq____"^15logseq____",[489,logseq____"^Jlogseq____",[],536870940]],[logseq____"^15logseq____",[489,logseq____"^Hlogseq____",220,536870940]],[logseq____"^15logseq____",[489,logseq____"^17logseq____",false,536870940]],[logseq____"^15logseq____",[489,logseq____"^;logseq____",logseq____"~u652fddf5-6c97-4e5b-b3c0-039282e36cb9logseq____",536870940]],[logseq____"^15logseq____",[490,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nlogseq____&\\\\neg\\\\left(\\\\forall x :\\\\exists y :\\\\forall w :\\\\forall u :\\\\exists z:p(x,y,z)\\\\rightarrow q(u,w)\\\\right)\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:(\\\\neg(\\\\exists y: \\\\forall w: \\\\forall u: \\\\exists z: p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:\\\\forall y:(\\\\neg(\\\\forall w: \\\\forall u:\\\\exists z: p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:\\\\forall y:\\\\exists w:(\\\\neg(\\\\forall u:\\\\exists z:p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x: \\\\forall y: \\\\exists w: \\\\exists u : \\\\forall z:\\\\neg(p(x,y,z)\\\\rightarrow 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logseq____",536870940]],[logseq____"^15logseq____",[491,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[491,logseq____"^Flogseq____",479,536870940]],[logseq____"^15logseq____",[491,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[491,logseq____"^Vlogseq____",479,536870940]],[logseq____"^15logseq____",[491,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[491,logseq____"^Ulogseq____",222,536870940]],[logseq____"^15logseq____",[491,logseq____"^Ulogseq____",225,536870940]],[logseq____"^15logseq____",[491,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[491,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[491,logseq____"^Ulogseq____",231,536870940]],[logseq____"^15logseq____",[491,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[491,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[491,logseq____"^Hlogseq____",222,536870940]],[logseq____"^15logseq____",[491,logseq____"^Hlogseq____",225,536870940]],[logseq____"^15logseq____",[491,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[491,logseq____"^;logseq____",logseq____"~u652fddf5-4616-429c-9e96-aa6d4d5c15b8logseq____",536870940]],[logseq____"^15logseq____",[492,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist wahr}\\n\\\\end{align}\\n[Beweis]([[Beweise]]):logseq____",536870940]],[logseq____"^15logseq____",[492,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[492,logseq____"^Flogseq____",475,536870940]],[logseq____"^15logseq____",[492,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[492,logseq____"^Vlogseq____",475,536870940]],[logseq____"^15logseq____",[492,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[492,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[492,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[492,logseq____"^Ulogseq____",231,536870940]],[logseq____"^15logseq____",[492,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[492,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[492,logseq____"^Hlogseq____",231,536870940]],[logseq____"^15logseq____",[492,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[492,logseq____"^;logseq____",logseq____"~u652fddf5-cceb-469d-86df-01485a830b73logseq____",536870940]],[logseq____"^15logseq____",[493,logseq____"^Qlogseq____",logseq____"Verschiedene [[Quantoren]] dürfen **nicht** vertauscht werdenlogseq____",536870940]],[logseq____"^15logseq____",[493,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[493,logseq____"^Flogseq____",496,536870940]],[logseq____"^15logseq____",[493,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[493,logseq____"^Vlogseq____",472,536870940]],[logseq____"^15logseq____",[493,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[493,logseq____"^Ulogseq____",228,536870940]],[logseq____"^15logseq____",[493,logseq____"^Ulogseq____",232,536870940]],[logseq____"^15logseq____",[493,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[493,logseq____"^Hlogseq____",232,536870940]],[logseq____"^15logseq____",[493,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[493,logseq____"^;logseq____",logseq____"~u652fddf5-60ff-4ae7-a4b9-1172f1298708logseq____",536870940]],[logseq____"^15logseq____",[494,logseq____"^Qlogseq____",logseq____"[[Aussageform]] (AF) über [Universen]([[Universen (Logik)]]) $U_1,\\\\cdots,U_n$:logseq____",536870940]],[logseq____"^15logseq____",[494,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[494,logseq____"^Flogseq____",478,536870940]],[logseq____"^15logseq____",[494,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[494,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[494,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[494,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[494,logseq____"^Ulogseq____",242,536870940]],[logseq____"^15logseq____",[494,logseq____"^Hlogseq____",238,536870940]],[logseq____"^15logseq____",[494,logseq____"^Hlogseq____",242,536870940]],[logseq____"^15logseq____",[494,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[494,logseq____"^;logseq____",logseq____"~u652fddf5-9843-4399-b026-0ea8d4ea98e4logseq____",536870940]],[logseq____"^15logseq____",[495,logseq____"^Qlogseq____",logseq____"## [[Regeln]]logseq____",536870940]],[logseq____"^15logseq____",[495,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[495,logseq____"^Flogseq____",471,536870940]],[logseq____"^15logseq____",[495,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[495,logseq____"^Vlogseq____",478,536870940]],[logseq____"^15logseq____",[495,logseq____"^Ulogseq____",220,536870940]],[logseq____"^15logseq____",[495,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[495,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[495,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870940]],[logseq____"^15logseq____",[495,logseq____"^Jlogseq____",[],536870940]],[logseq____"^15logseq____",[495,logseq____"^Hlogseq____",220,536870940]],[logseq____"^15logseq____",[495,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[495,logseq____"^;logseq____",logseq____"~u652fddf5-6b41-469f-930c-924af30067e2logseq____",536870940]],[logseq____"^15logseq____",[496,logseq____"^Qlogseq____",logseq____"wobei $\\\\pi$ eine \\logseq____"[[Permutation]]\\logseq____" der Zahl $(1,2,\\\\cdots,n)$ 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von quantoren ([Beispiel]([[Beispiele]])):logseq____",536870940]],[logseq____"^15logseq____",[497,logseq____"^Ologseq____",logseq____"^16logseq____",536870940]],[logseq____"^15logseq____",[497,logseq____"^Flogseq____",501,536870940]],[logseq____"^15logseq____",[497,logseq____"^Xlogseq____",227,536870940]],[logseq____"^15logseq____",[497,logseq____"^Vlogseq____",495,536870940]],[logseq____"^15logseq____",[497,logseq____"^Ulogseq____",39,536870940]],[logseq____"^15logseq____",[497,logseq____"^Ulogseq____",220,536870940]],[logseq____"^15logseq____",[497,logseq____"^Ulogseq____",227,536870940]],[logseq____"^15logseq____",[497,logseq____"^Ulogseq____",238,536870940]],[logseq____"^15logseq____",[497,logseq____"^Hlogseq____",39,536870940]],[logseq____"^15logseq____",[497,logseq____"^17logseq____",true,536870940]],[logseq____"^15logseq____",[497,logseq____"^;logseq____",logseq____"~u652fddf5-f571-4b08-b01e-6a894f7c2e69logseq____",536870940]],[logseq____"^15logseq____",[498,logseq____"^Qlogseq____",logseq____"$\\\\exists 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