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logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[54,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[54,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[54,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[54,logseq____"^;logseq____",logseq____"~u65297d30-1eee-4048-946a-bacfcd6b8a5alogseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Qlogseq____",logseq____"Einsetzen, um zu überprüfen, ob man eine Fallunterscheidung brauchtlogseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Flogseq____",92,536870919]],[logseq____"^15logseq____",[55,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^Vlogseq____",63,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[55,logseq____"^;logseq____",logseq____"~u65297d30-3f4a-46df-b3d1-0b5329936522logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Qlogseq____",logseq____"[[Wolframalpha]]logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Flogseq____",78,536870919]],[logseq____"^15logseq____",[56,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Vlogseq____",78,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[56,logseq____"^Hlogseq____",46,536870919]],[logseq____"^15logseq____",[56,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[56,logseq____"^;logseq____",logseq____"~u65297d30-892a-4eca-9bf2-9f76e788db9alogseq____",536870919]],[logseq____"^15logseq____",[57,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____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Flogseq____",101,536870919]],[logseq____"^15logseq____",[57,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Vlogseq____",101,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[57,logseq____"^;logseq____",logseq____"~u65297d30-bda3-4c4e-a4a4-2711ec3d3aeclogseq____",536870919]],[logseq____"^15logseq____",[58,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 x)^5logseq____&=2+20x+40x^2\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Flogseq____",76,536870919]],[logseq____"^15logseq____",[58,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[58,logseq____"^;logseq____",logseq____"~u65297d30-dbe8-40c4-a695-8fdc9ea772d0logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Flogseq____",100,536870919]],[logseq____"^15logseq____",[59,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Vlogseq____",79,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[59,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[59,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[59,logseq____"^;logseq____",logseq____"~u65297d30-5ccd-4b36-af82-fbe91ad02f29logseq____",536870919]],[logseq____"^15logseq____",[60,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____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Flogseq____",66,536870919]],[logseq____"^15logseq____",[60,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Vlogseq____",66,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[60,logseq____"^;logseq____",logseq____"~u65297d30-2210-4b08-a7c6-43e86c095fa2logseq____",536870919]],[logseq____"^15logseq____",[61,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____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Flogseq____",91,536870919]],[logseq____"^15logseq____",[61,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^Vlogseq____",91,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[61,logseq____"^;logseq____",logseq____"~u65297d30-71bc-4e74-84a2-884f4e968aa9logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Qlogseq____",logseq____"Das [[Pascalsche Dreieck]] lässt sich auch mit den entsprechenden Binomialkoeffizienten darstellenlogseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Flogseq____",64,536870919]],[logseq____"^15logseq____",[62,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^Vlogseq____",64,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",38,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^Hlogseq____",38,536870919]],[logseq____"^15logseq____",[62,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[62,logseq____"^;logseq____",logseq____"~u65297d30-991c-4ce9-9927-23a5cc7fb459logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Qlogseq____",logseq____"# [[Wurzelgleichung]]logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Flogseq____",85,536870919]],[logseq____"^15logseq____",[63,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^Vlogseq____",67,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[63,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[63,logseq____"^Hlogseq____",30,536870919]],[logseq____"^15logseq____",[63,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[63,logseq____"^;logseq____",logseq____"~u65297d30-2d1b-4cb2-8705-69b3f059a0d2logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Qlogseq____",logseq____"## [[Binomialkoeffizient]]logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Flogseq____",76,536870919]],[logseq____"^15logseq____",[64,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^Vlogseq____",101,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[64,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[64,logseq____"^Hlogseq____",32,536870919]],[logseq____"^15logseq____",[64,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[64,logseq____"^;logseq____",logseq____"~u65297d30-3144-42e9-84b9-745f87704757logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(a+b)^3(a^2+b^2)^3(a-b)^3logseq____&=(a^2+b^2)^3\\\\left((a+b)(a-b)\\\\right)^3\\\\notag\\\\\\\\\\nlogseq____&=(a^2+b^2)^3(a^2-b^2)^3\\\\notag\\\\\\\\\\nlogseq____&=\\\\left((a^2+b^2)(a^2-b^2)\\\\right)^3\\\\notag\\\\\\\\\\nlogseq____&=(a^4-b^4)^3\\\\\\\\\\n\\\\frac{a-b}{(a+b)^{-1}}logseq____&=(a-b)(a+b)\\\\notag\\\\\\\\\\nlogseq____&=a^2-b^2\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Flogseq____",86,536870919]],[logseq____"^15logseq____",[65,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[65,logseq____"^;logseq____",logseq____"~u65297d30-8a73-4abf-bb9c-a754d08428c2logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Trigonometrie]]logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Flogseq____",94,536870919]],[logseq____"^15logseq____",[66,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[66,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[66,logseq____"^Hlogseq____",36,536870919]],[logseq____"^15logseq____",[66,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[66,logseq____"^;logseq____",logseq____"~u65297d30-acea-43e4-b788-40cdf2958986logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Qlogseq____",logseq____"# [[Potenzen]] und [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Flogseq____",51,536870919]],[logseq____"^15logseq____",[67,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[67,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[67,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[67,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[67,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[67,logseq____"^;logseq____",logseq____"~u65297d30-51ef-4e58-8a48-cfef85daf158logseq____",536870919]],[logseq____"^15logseq____",[68,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____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Flogseq____",95,536870919]],[logseq____"^15logseq____",[68,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Vlogseq____",95,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[68,logseq____"^;logseq____",logseq____"~u65297d30-b4e9-4910-a931-9069d1416905logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn ergibtlogseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Flogseq____",53,536870919]],[logseq____"^15logseq____",[69,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^Vlogseq____",48,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[69,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[69,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[69,logseq____"^;logseq____",logseq____"~u65297d30-8b7d-4122-adb3-5beaedb597f0logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Qlogseq____",logseq____"# [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Flogseq____",49,536870919]],[logseq____"^15logseq____",[70,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^Vlogseq____",67,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[70,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[70,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[70,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[70,logseq____"^;logseq____",logseq____"~u65297d30-5002-4155-9b59-fef118708325logseq____",536870919]],[logseq____"^15logseq____",[71,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____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Flogseq____",47,536870919]],[logseq____"^15logseq____",[71,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Vlogseq____",51,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[71,logseq____"^;logseq____",logseq____"~u65297d30-8102-4a10-9edb-4ef7ef5c5e50logseq____",536870919]],[logseq____"^15logseq____",[72,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____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Flogseq____",89,536870919]],[logseq____"^15logseq____",[72,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Vlogseq____",89,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",26,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[72,logseq____"^Hlogseq____",26,536870919]],[logseq____"^15logseq____",[72,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[72,logseq____"^;logseq____",logseq____"~u65297d30-5f9a-4cf5-8ebb-e0329c791eeblogseq____",536870919]],[logseq____"^15logseq____",[73,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____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Flogseq____",68,536870919]],[logseq____"^15logseq____",[73,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^Vlogseq____",95,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[73,logseq____"^;logseq____",logseq____"~u65297d30-628b-49b5-8ad0-ff74e9d28587logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Qlogseq____",logseq____"logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Flogseq____",80,536870919]],[logseq____"^15logseq____",[74,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[74,logseq____"^;logseq____",logseq____"~u65297d30-4ab4-49b2-b3cd-3a82bcdaf426logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Flogseq____",102,536870919]],[logseq____"^15logseq____",[75,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^Vlogseq____",102,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[75,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[75,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[75,logseq____"^;logseq____",logseq____"~u65297d30-fb9b-4278-bbde-435a21537346logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Flogseq____",57,536870919]],[logseq____"^15logseq____",[76,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Vlogseq____",101,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[76,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[76,logseq____"^;logseq____",logseq____"~u65297d30-704a-46e7-a06b-e30f8b0e4ae4logseq____",536870919]],[logseq____"^15logseq____",[77,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____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Flogseq____",61,536870919]],[logseq____"^15logseq____",[77,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^Vlogseq____",91,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[77,logseq____"^;logseq____",logseq____"~u65297d30-57cb-4114-972e-e9ed589dd448logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Qlogseq____",logseq____"# [[Links]]logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Flogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[78,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[78,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[78,logseq____"^Hlogseq____",44,536870919]],[logseq____"^15logseq____",[78,logseq____"^17logseq____",false,536870919]],[logseq____"^15logseq____",[78,logseq____"^;logseq____",logseq____"~u65297d30-692a-4f24-89de-73c0975148a8logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Qlogseq____",logseq____"## [[Rationalmachen]] des [[Nenner]]slogseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Flogseq____",83,536870919]],[logseq____"^15logseq____",[79,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Vlogseq____",70,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[79,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[79,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[79,logseq____"^Hlogseq____",39,536870919]],[logseq____"^15logseq____",[79,logseq____"^Hlogseq____",45,536870919]],[logseq____"^15logseq____",[79,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[79,logseq____"^;logseq____",logseq____"~u65297d30-dcf1-4d0a-9ace-87e86f49910blogseq____",536870919]],[logseq____"^15logseq____",[80,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____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Flogseq____",54,536870919]],[logseq____"^15logseq____",[80,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",33,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^Hlogseq____",33,536870919]],[logseq____"^15logseq____",[80,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[80,logseq____"^;logseq____",logseq____"~u65297d30-e161-4bf9-9901-d3ee8f9f2badlogseq____",536870919]],[logseq____"^15logseq____",[81,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____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Flogseq____",59,536870919]],[logseq____"^15logseq____",[81,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^Vlogseq____",59,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[81,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[81,logseq____"^;logseq____",logseq____"~u65297d30-b110-4b2e-bbd4-4ee634a4aac4logseq____",536870919]],[logseq____"^15logseq____",[82,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____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Flogseq____",86,536870919]],[logseq____"^15logseq____",[82,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^Vlogseq____",49,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[82,logseq____"^;logseq____",logseq____"~u65297d30-6905-4fde-b81e-37bfeed77756logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Qlogseq____",logseq____"$\\\\text{Allgemein}$\\n\\\\begin{align}\\n|y|logseq____&=\\\\begin{cases}\\ny logseq____&\\\\text{ falls } y\\\\ge0\\\\\\\\\\n-y logseq____&\\\\text{ falls } ylogseq____<0\\\\\\\\\\n\\\\end{cases}\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Flogseq____",48,536870919]],[logseq____"^15logseq____",[83,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^Vlogseq____",70,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[83,logseq____"^;logseq____",logseq____"~u65297d30-115e-4ec7-86a4-b1afff098017logseq____",536870919]],[logseq____"^15logseq____",[84,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____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Flogseq____",49,536870919]],[logseq____"^15logseq____",[84,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^Vlogseq____",49,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[84,logseq____"^;logseq____",logseq____"~u65297d30-8053-4066-845e-3d12ec885762logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Qlogseq____",logseq____"# [[Quadratische Gleichungen]]logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Flogseq____",70,536870919]],[logseq____"^15logseq____",[85,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^Vlogseq____",67,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[85,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[85,logseq____"^Hlogseq____",29,536870919]],[logseq____"^15logseq____",[85,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[85,logseq____"^;logseq____",logseq____"~u65297d30-2dae-4a5b-98d0-f123079ef4d9logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Flogseq____",84,536870919]],[logseq____"^15logseq____",[86,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Vlogseq____",49,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[86,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[86,logseq____"^;logseq____",logseq____"~u65297d30-01cf-408e-9806-80a7904a3f29logseq____",536870919]],[logseq____"^15logseq____",[87,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____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Flogseq____",103,536870919]],[logseq____"^15logseq____",[87,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^Vlogseq____",70,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[87,logseq____"^;logseq____",logseq____"~u65297d30-299a-458d-8aa9-e57e44cb3d79logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Qlogseq____",logseq____"https://www.wolframalpha.comlogseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Flogseq____",56,536870919]],[logseq____"^15logseq____",[88,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Vlogseq____",56,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[88,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[88,logseq____"^;logseq____",logseq____"~u65297d30-f1f3-4cde-b49d-81714c0edb73logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Qlogseq____",logseq____"## [[pq-Formel]]logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Flogseq____",50,536870919]],[logseq____"^15logseq____",[89,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Vlogseq____",85,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[89,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[89,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[89,logseq____"^Hlogseq____",43,536870919]],[logseq____"^15logseq____",[89,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[89,logseq____"^;logseq____",logseq____"~u65297d30-638f-4173-abc6-da7dc5c803f5logseq____",536870919]],[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____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Flogseq____",102,536870919]],[logseq____"^15logseq____",[90,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Vlogseq____",89,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[90,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[90,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[90,logseq____"^;logseq____",logseq____"~u65297d30-587f-4518-b3d2-e060b1e60e06logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Logarithmen]]logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Flogseq____",54,536870919]],[logseq____"^15logseq____",[91,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[91,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[91,logseq____"^Hlogseq____",31,536870919]],[logseq____"^15logseq____",[91,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[91,logseq____"^;logseq____",logseq____"~u65297d30-43e2-4b5d-be74-d94d72979c8elogseq____",536870919]],[logseq____"^15logseq____",[92,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____",536870919]],[logseq____"^15logseq____",[92,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[92,logseq____"^Flogseq____",63,536870919]],[logseq____"^15logseq____",[92,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^Vlogseq____",63,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[92,logseq____"^;logseq____",logseq____"~u65297d30-5a58-44f1-ab62-90f61435dabdlogseq____",536870919]],[logseq____"^15logseq____",[93,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____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Flogseq____",60,536870919]],[logseq____"^15logseq____",[93,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^Vlogseq____",66,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[93,logseq____"^;logseq____",logseq____"~u65297d30-2744-480d-9bd1-31ec23aaa1fflogseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Qlogseq____",logseq____"# Was ist Größer?logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Flogseq____",95,536870919]],[logseq____"^15logseq____",[94,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[94,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[94,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[94,logseq____"^;logseq____",logseq____"~u65297d30-a06f-480e-888e-a4fe9666a615logseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Flogseq____",91,536870919]],[logseq____"^15logseq____",[95,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[95,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[95,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[95,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[95,logseq____"^;logseq____",logseq____"~u65297d30-4f58-44f5-9e28-d33aa25c2007logseq____",536870919]],[logseq____"^15logseq____",[96,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____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Flogseq____",94,536870919]],[logseq____"^15logseq____",[96,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Vlogseq____",94,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[96,logseq____"^;logseq____",logseq____"~u65297d30-8003-48b6-af09-d3337d14e2d4logseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Qlogseq____",logseq____"#FIXME $(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5=2+20x+40x^2$ ist Falschlogseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Flogseq____",58,536870919]],[logseq____"^15logseq____",[97,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[97,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[97,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[97,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[97,logseq____"^;logseq____",logseq____"~u65297d30-8c13-4337-8bc5-0ac153ec1538logseq____",536870919]],[logseq____"^15logseq____",[98,logseq____"^Qlogseq____",logseq____"cosa 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{2\\\\choose 2}\\n\\\\\\\\\\n{3\\\\choose 0} {3\\\\choose 1} {3\\\\choose 2} {3\\\\choose 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[[Kreis]]logseq____",536870920]],[logseq____"^15logseq____",[135,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[135,logseq____"^Flogseq____",122,536870920]],[logseq____"^15logseq____",[135,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[135,logseq____"^Vlogseq____",145,536870920]],[logseq____"^15logseq____",[135,logseq____"^Ulogseq____",106,536870920]],[logseq____"^15logseq____",[135,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[135,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[135,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870920]],[logseq____"^15logseq____",[135,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[135,logseq____"^Hlogseq____",106,536870920]],[logseq____"^15logseq____",[135,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[135,logseq____"^;logseq____",logseq____"~u65297d30-7448-4d54-90cb-261e1d1c9d8dlogseq____",536870920]],[logseq____"^15logseq____",[136,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 1a}\\\\quad x\\\\ge -2$\\n\\\\begin{align}\\n|x+2|logseq____&=x+2\\\\\\\\\\nx-3logseq____&\\\\le x+2 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aller reellen Zahlen zwischen 1 und 4 inklusive 1 und 4\\n\\\\begin{align}\\\\left\\\\{x\\\\in\\\\Reals\\\\middle|-1\\\\le x\\\\le4\\\\right\\\\}=:\\\\underbrace{[-1,4]}_\\\\text{Abgeschlossenes 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aller rellen Zahlen zwischen 4 und $\\\\infty$ ohne 4 und $\\\\infty$\\n\\\\begin{align}\\\\left\\\\{x\\\\in\\\\Reals\\\\middle|xlogseq____>4\\\\right\\\\}=:(4,\\\\infty)\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[141,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[141,logseq____"^Flogseq____",131,536870920]],[logseq____"^15logseq____",[141,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[141,logseq____"^Vlogseq____",150,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[141,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[141,logseq____"^;logseq____",logseq____"~u65297d30-2df9-4ad9-9298-0b15e409a451logseq____",536870920]],[logseq____"^15logseq____",[142,logseq____"^Qlogseq____",logseq____"Bei Multiplikation oder Division mit einem Betrag einer negativen Zahl nicht Anwendbar, da ein [[Betrag]] immer Positiv istlogseq____",536870920]],[logseq____"^15logseq____",[142,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[142,logseq____"^Flogseq____",115,536870920]],[logseq____"^15logseq____",[142,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[142,logseq____"^Vlogseq____",115,536870920]],[logseq____"^15logseq____",[142,logseq____"^Ulogseq____",108,536870920]],[logseq____"^15logseq____",[142,logseq____"^Ulogseq____",111,536870920]],[logseq____"^15logseq____",[142,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[142,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[142,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[142,logseq____"^Hlogseq____",111,536870920]],[logseq____"^15logseq____",[142,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[142,logseq____"^;logseq____",logseq____"~u65297d30-a574-470f-9eaf-75c428da6b01logseq____",536870920]],[logseq____"^15logseq____",[143,logseq____"^Qlogseq____",logseq____"### Lösung mit [[Fallunterscheidung]]logseq____",536870920]],[logseq____"^15logseq____",[143,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[143,logseq____"^Flogseq____",120,536870920]],[logseq____"^15logseq____",[143,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[143,logseq____"^Vlogseq____",120,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",107,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",111,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[143,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870920]],[logseq____"^15logseq____",[143,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[143,logseq____"^Hlogseq____",107,536870920]],[logseq____"^15logseq____",[143,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[143,logseq____"^;logseq____",logseq____"~u65297d30-1909-40fa-a3c1-1a7c868862dblogseq____",536870920]],[logseq____"^15logseq____",[144,logseq____"^Qlogseq____",logseq____"$$L=\\\\left\\\\{\\\\underbrace{x\\\\in\\\\Reals}_{0\\\\notin L}\\\\:\\\\middle|\\\\:\\\\underbrace{|x-3|\\\\le|x+2}_{1\\\\in L}\\\\right\\\\}$$\\n$0\\\\notin L\\\\rightarrow\\\\text{0 steht nicht in Lösungsmenge}$\\n$1\\\\in L\\\\rightarrow\\\\text{1 gehört zu Lösungsmenge}$logseq____",536870920]],[logseq____"^15logseq____",[144,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[144,logseq____"^Flogseq____",146,536870920]],[logseq____"^15logseq____",[144,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[144,logseq____"^Vlogseq____",146,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",111,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[144,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[144,logseq____"^;logseq____",logseq____"~u65297d30-1b8a-4535-91bf-2ac6681f116elogseq____",536870920]],[logseq____"^15logseq____",[145,logseq____"^Qlogseq____",logseq____"# [[Mengen]]logseq____",536870920]],[logseq____"^15logseq____",[145,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[145,logseq____"^Flogseq____",112,536870920]],[logseq____"^15logseq____",[145,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[145,logseq____"^Vlogseq____",112,536870920]],[logseq____"^15logseq____",[145,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[145,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[145,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870920]],[logseq____"^15logseq____",[145,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[145,logseq____"^Hlogseq____",114,536870920]],[logseq____"^15logseq____",[145,logseq____"^17logseq____",false,536870920]],[logseq____"^15logseq____",[145,logseq____"^;logseq____",logseq____"~u65297d30-8ec5-43da-9dd3-1e250b8937cflogseq____",536870920]],[logseq____"^15logseq____",[146,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870920]],[logseq____"^15logseq____",[146,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[146,logseq____"^Flogseq____",115,536870920]],[logseq____"^15logseq____",[146,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[146,logseq____"^Vlogseq____",133,536870920]],[logseq____"^15logseq____",[146,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[146,logseq____"^Ulogseq____",111,536870920]],[logseq____"^15logseq____",[146,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[146,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[146,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[146,logseq____"^Hlogseq____",35,536870920]],[logseq____"^15logseq____",[146,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[146,logseq____"^;logseq____",logseq____"~u65297d30-6320-4da6-94b5-697c77fecddflogseq____",536870920]],[logseq____"^15logseq____",[147,logseq____"^Qlogseq____",logseq____"## 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2011logseq____",536870922]],[logseq____"^15logseq____",[167,logseq____"^Blogseq____",1697217841010,536870922]],[logseq____"^15logseq____",[167,logseq____"^;logseq____",logseq____"~u65297d31-6fde-485a-af42-36c5bae19e29logseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^Klogseq____",1697217841024,536870922]],[logseq____"^15logseq____",[168,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ylogseq____",logseq____"tantologielogseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^11logseq____",logseq____"Tantologielogseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^Blogseq____",1697217841024,536870922]],[logseq____"^15logseq____",[168,logseq____"^;logseq____",logseq____"~u65297d31-081c-4892-9150-1552cd2ad350logseq____",536870922]],[logseq____"^15logseq____",[169,logseq____"^Klogseq____",1697217841023,536870922]],[logseq____"^15logseq____",[169,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ylogseq____",logseq____"kontradiktionlogseq____",536870922]],[logseq____"^15logseq____",[169,logseq____"^11logseq____",logseq____"Kontradiktionlogseq____",536870922]],[logseq____"^15logseq____",[169,logseq____"^Blogseq____",1697217841023,536870922]],[logseq____"^15logseq____",[169,logseq____"^;logseq____",logseq____"~u65297d31-0eea-4da1-b35d-bb1be4896b7clogseq____",536870922]],[logseq____"^15logseq____",[170,logseq____"^Klogseq____",1697217841020,536870922]],[logseq____"^15logseq____",[170,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[170,logseq____"^Ylogseq____",logseq____"variablenlogseq____",536870922]],[logseq____"^15logseq____",[170,logseq____"^11logseq____",logseq____"Variablenlogseq____",536870922]],[logseq____"^15logseq____",[170,logseq____"^Blogseq____",1697217841020,536870922]],[logseq____"^15logseq____",[170,logseq____"^;logseq____",logseq____"~u65297d31-3e6c-4990-96d0-cc06931e9713logseq____",536870923]],[logseq____"^15logseq____",[171,logseq____"^Klogseq____",1697217841015,536870922]],[logseq____"^15logseq____",[171,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[171,logseq____"^Ylogseq____",logseq____"semantiklogseq____",536870922]],[logseq____"^15logseq____",[171,logseq____"^11logseq____",logseq____"Semantiklogseq____",536870922]],[logseq____"^15logseq____",[171,logseq____"^Blogseq____",1697217841015,536870922]],[logseq____"^15logseq____",[171,logseq____"^;logseq____",logseq____"~u65297d31-2dea-4cff-a133-467c8f1c026alogseq____",536870922]],[logseq____"^15logseq____",[172,logseq____"^Klogseq____",1697217841024,536870922]],[logseq____"^15logseq____",[172,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[172,logseq____"^Ylogseq____",logseq____"logisch äquivalentlogseq____",536870922]],[logseq____"^15logseq____",[172,logseq____"^11logseq____",logseq____"logisch äquivalentlogseq____",536870922]],[logseq____"^15logseq____",[172,logseq____"^Blogseq____",1697217841024,536870922]],[logseq____"^15logseq____",[172,logseq____"^;logseq____",logseq____"~u65297d31-26ed-4fff-89bf-04108dab52cclogseq____",536870922]],[logseq____"^15logseq____",[173,logseq____"^Klogseq____",1697217841020,536870922]],[logseq____"^15logseq____",[173,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[173,logseq____"^Ylogseq____",logseq____"tabellenregelnlogseq____",536870922]],[logseq____"^15logseq____",[173,logseq____"^11logseq____",logseq____"Tabellenregelnlogseq____",536870922]],[logseq____"^15logseq____",[173,logseq____"^Blogseq____",1697217841020,536870922]],[logseq____"^15logseq____",[173,logseq____"^;logseq____",logseq____"~u65297d31-4f1b-468f-b4e7-b8d55a4c15e8logseq____",536870922]],[logseq____"^15logseq____",[174,logseq____"^Klogseq____",1697217841013,536870922]],[logseq____"^15logseq____",[174,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[174,logseq____"^Ylogseq____",logseq____"aussagenlogseq____",536870922]],[logseq____"^15logseq____",[174,logseq____"^11logseq____",logseq____"Aussagenlogseq____",536870922]],[logseq____"^15logseq____",[174,logseq____"^Blogseq____",1697217841013,536870922]],[logseq____"^15logseq____",[174,logseq____"^;logseq____",logseq____"~u65297d31-e0f7-447e-926c-450d56686717logseq____",536870923]],[logseq____"^15logseq____",[175,logseq____"^Klogseq____",1697217841019,536870922]],[logseq____"^15logseq____",[175,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[175,logseq____"^Ylogseq____",logseq____"verknüpfungenlogseq____",536870922]],[logseq____"^15logseq____",[175,logseq____"^11logseq____",logseq____"Verknüpfungenlogseq____",536870922]],[logseq____"^15logseq____",[175,logseq____"^Blogseq____",1697217841019,536870922]],[logseq____"^15logseq____",[175,logseq____"^;logseq____",logseq____"~u65297d31-406c-4899-8f4e-ace909c7d6c4logseq____",536870922]],[logseq____"^15logseq____",[176,logseq____"^Klogseq____",1697217841020,536870922]],[logseq____"^15logseq____",[176,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[176,logseq____"^Ylogseq____",logseq____"aussageformellogseq____",536870922]],[logseq____"^15logseq____",[176,logseq____"^11logseq____",logseq____"Aussageformellogseq____",536870922]],[logseq____"^15logseq____",[176,logseq____"^Blogseq____",1697217841020,536870922]],[logseq____"^15logseq____",[176,logseq____"^;logseq____",logseq____"~u65297d31-7c38-466a-ad64-f81288cc2a2alogseq____",536870922]],[logseq____"^15logseq____",[177,logseq____"^Klogseq____",1697217841016,536870922]],[logseq____"^15logseq____",[177,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[177,logseq____"^Ylogseq____",logseq____"syntaxlogseq____",536870922]],[logseq____"^15logseq____",[177,logseq____"^11logseq____",logseq____"Syntaxlogseq____",536870922]],[logseq____"^15logseq____",[177,logseq____"^Blogseq____",1697217841016,536870922]],[logseq____"^15logseq____",[177,logseq____"^;logseq____",logseq____"~u65297d31-5679-4e8f-b151-d751b87b1210logseq____",536870922]],[logseq____"^15logseq____",[178,logseq____"^Klogseq____",1697217841013,536870922]],[logseq____"^15logseq____",[178,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[178,logseq____"^Ylogseq____",logseq____"verknüpfung von aussagenlogseq____",536870922]],[logseq____"^15logseq____",[178,logseq____"^11logseq____",logseq____"Verknüpfung von Aussagenlogseq____",536870922]],[logseq____"^15logseq____",[178,logseq____"^Blogseq____",1697217841013,536870922]],[logseq____"^15logseq____",[178,logseq____"^;logseq____",logseq____"~u65297d31-8be3-424f-9088-0232646247dflogseq____",536870922]],[logseq____"^15logseq____",[179,logseq____"^Klogseq____",1697217841023,536870922]],[logseq____"^15logseq____",[179,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[179,logseq____"^Ylogseq____",logseq____"schreibweiselogseq____",536870922]],[logseq____"^15logseq____",[179,logseq____"^11logseq____",logseq____"Schreibweiselogseq____",536870922]],[logseq____"^15logseq____",[179,logseq____"^Blogseq____",1697217841023,536870922]],[logseq____"^15logseq____",[179,logseq____"^;logseq____",logseq____"~u65297d31-544f-45c1-a9c2-80c5edee4c31logseq____",536870922]],[logseq____"^15logseq____",[180,logseq____"^Klogseq____",1697217841021,536870922]],[logseq____"^15logseq____",[180,logseq____"^@logseq____",false,536870922]],[logseq____"^15logseq____",[180,logseq____"^Ylogseq____",logseq____"belegunglogseq____",536870922]],[logseq____"^15logseq____",[180,logseq____"^11logseq____",logseq____"Belegunglogseq____",536870922]],[logseq____"^15logseq____",[180,logseq____"^Blogseq____",1697217841021,536870922]],[logseq____"^15logseq____",[180,logseq____"^;logseq____",logseq____"~u65297d31-397c-4be6-8568-7a038de6c185logseq____",536870922]],[logseq____"^15logseq____",[181,logseq____"^Qlogseq____",logseq____"# Büchertiplogseq____",536870922]],[logseq____"^15logseq____",[181,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[181,logseq____"^Flogseq____",159,536870922]],[logseq____"^15logseq____",[181,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[181,logseq____"^Vlogseq____",159,536870922]],[logseq____"^15logseq____",[181,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[181,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[181,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[181,logseq____"^17logseq____",false,536870922]],[logseq____"^15logseq____",[181,logseq____"^;logseq____",logseq____"~u65297d30-5d82-4ba5-8a0a-d16099a8852blogseq____",536870922]],[logseq____"^15logseq____",[182,logseq____"^Qlogseq____",logseq____"Alternativ: $p\\\\equiv q$, falls $p\\\\leftrightarrow q$ [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[182,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[182,logseq____"^Flogseq____",188,536870922]],[logseq____"^15logseq____",[182,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[182,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[182,logseq____"^Hlogseq____",168,536870922]],[logseq____"^15logseq____",[182,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[182,logseq____"^;logseq____",logseq____"~u65297d30-a519-4aa7-bcb2-ccdd2840ecc4logseq____",536870922]],[logseq____"^15logseq____",[183,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____",536870922]],[logseq____"^15logseq____",[183,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[183,logseq____"^Flogseq____",193,536870922]],[logseq____"^15logseq____",[183,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[183,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[183,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[183,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[183,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[183,logseq____"^Hlogseq____",174,536870922]],[logseq____"^15logseq____",[183,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[183,logseq____"^;logseq____",logseq____"~u65297d30-d2e5-4a8f-a781-f488c35506b7logseq____",536870922]],[logseq____"^15logseq____",[184,logseq____"^Qlogseq____",logseq____"Formel, die konstant $f$ istlogseq____",536870922]],[logseq____"^15logseq____",[184,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[184,logseq____"^Flogseq____",207,536870922]],[logseq____"^15logseq____",[184,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[184,logseq____"^Vlogseq____",207,536870922]],[logseq____"^15logseq____",[184,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[184,logseq____"^Ulogseq____",169,536870922]],[logseq____"^15logseq____",[184,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[184,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[184,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[184,logseq____"^;logseq____",logseq____"~u65297d30-fd18-4adb-a3a4-39ff9fabc9d7logseq____",536870922]],[logseq____"^15logseq____",[185,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____",536870922]],[logseq____"^15logseq____",[185,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[185,logseq____"^Flogseq____",206,536870922]],[logseq____"^15logseq____",[185,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[185,logseq____"^Vlogseq____",199,536870922]],[logseq____"^15logseq____",[185,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[185,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[185,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[185,logseq____"^;logseq____",logseq____"~u65297d30-e130-4976-8306-969641c99a70logseq____",536870922]],[logseq____"^15logseq____",[186,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____",536870922]],[logseq____"^15logseq____",[186,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[186,logseq____"^Flogseq____",196,536870922]],[logseq____"^15logseq____",[186,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[186,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",166,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",171,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",173,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",177,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[186,logseq____"^Hlogseq____",166,536870922]],[logseq____"^15logseq____",[186,logseq____"^Hlogseq____",171,536870922]],[logseq____"^15logseq____",[186,logseq____"^Hlogseq____",173,536870922]],[logseq____"^15logseq____",[186,logseq____"^Hlogseq____",177,536870922]],[logseq____"^15logseq____",[186,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[186,logseq____"^;logseq____",logseq____"~u65297d30-ce20-46cd-b489-85950833f361logseq____",536870922]],[logseq____"^15logseq____",[187,logseq____"^Qlogseq____",logseq____"### [[Belegung]] (der [[Variablen]])logseq____",536870922]],[logseq____"^15logseq____",[187,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[187,logseq____"^Flogseq____",201,536870922]],[logseq____"^15logseq____",[187,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[187,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",170,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[187,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[187,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[187,logseq____"^Hlogseq____",170,536870922]],[logseq____"^15logseq____",[187,logseq____"^Hlogseq____",180,536870922]],[logseq____"^15logseq____",[187,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[187,logseq____"^;logseq____",logseq____"~u65297d30-9a07-44bc-aad6-bec57367c422logseq____",536870922]],[logseq____"^15logseq____",[188,logseq____"^Qlogseq____",logseq____"Formeln $p,q$ [[logisch äquivalent]], wenn sie den gleichen [[Wahrheitswerteverlauf]] haben.\\n[[Schreibweise]]: $p\\\\equiv q$logseq____",536870922]],[logseq____"^15logseq____",[188,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[188,logseq____"^Flogseq____",207,536870922]],[logseq____"^15logseq____",[188,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[188,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",162,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",172,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",179,536870922]],[logseq____"^15logseq____",[188,logseq____"^Hlogseq____",162,536870922]],[logseq____"^15logseq____",[188,logseq____"^Hlogseq____",172,536870922]],[logseq____"^15logseq____",[188,logseq____"^Hlogseq____",179,536870922]],[logseq____"^15logseq____",[188,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[188,logseq____"^;logseq____",logseq____"~u65297d30-0559-48c5-9dd3-4a66041c7030logseq____",536870922]],[logseq____"^15logseq____",[189,logseq____"^Qlogseq____",logseq____"Weitere 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mögliche [[Belegung]] wird ein [[Wahrheitswert]] der [[Formel]] zugeordnet (nach 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[[Verknüpfung von Aussagen]]logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Flogseq____",185,536870922]],[logseq____"^15logseq____",[193,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[193,logseq____"^Vlogseq____",199,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[193,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870922]],[logseq____"^15logseq____",[193,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[193,logseq____"^Hlogseq____",178,536870922]],[logseq____"^15logseq____",[193,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[193,logseq____"^;logseq____",logseq____"~u65297d30-3b14-4b7d-85d6-bebc999c4ca7logseq____",536870922]],[logseq____"^15logseq____",[194,logseq____"^Qlogseq____",logseq____"### [[Wahrheitswerteverlauf]] einer [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[194,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[194,logseq____"^Flogseq____",187,536870922]],[logseq____"^15logseq____",[194,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[194,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",162,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[194,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[194,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[194,logseq____"^Hlogseq____",162,536870922]],[logseq____"^15logseq____",[194,logseq____"^Hlogseq____",176,536870922]],[logseq____"^15logseq____",[194,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[194,logseq____"^;logseq____",logseq____"~u65297d30-d4a7-408e-85e2-061f3dd8fecclogseq____",536870922]],[logseq____"^15logseq____",[195,logseq____"^Qlogseq____",logseq____"### [[Aussagenlogische 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logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[195,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[195,logseq____"^Hlogseq____",163,536870922]],[logseq____"^15logseq____",[195,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[195,logseq____"^;logseq____",logseq____"~u65297d30-09a1-4563-8098-17a50741ae29logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Flogseq____",183,536870922]],[logseq____"^15logseq____",[196,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[196,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[196,logseq____"^Hlogseq____",35,536870922]],[logseq____"^15logseq____",[196,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[196,logseq____"^;logseq____",logseq____"~u65297d30-866f-48b0-808b-887ef07c3487logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Qlogseq____",logseq____"[[Wikipedia]]logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Flogseq____",200,536870922]],[logseq____"^15logseq____",[197,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[197,logseq____"^Vlogseq____",181,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",164,536870922]],[logseq____"^15logseq____",[197,logseq____"^Hlogseq____",164,536870922]],[logseq____"^15logseq____",[197,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[197,logseq____"^;logseq____",logseq____"~u65297d30-6449-494e-948f-31ac76baa981logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Qlogseq____",logseq____"### [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Flogseq____",194,536870922]],[logseq____"^15logseq____",[198,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[198,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[198,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[198,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[198,logseq____"^Hlogseq____",168,536870922]],[logseq____"^15logseq____",[198,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[198,logseq____"^;logseq____",logseq____"~u65297d30-04a2-48ba-9a6c-53d6364a98eblogseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Qlogseq____",logseq____"# 1. [[Aussagen]]logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Flogseq____",181,536870922]],[logseq____"^15logseq____",[199,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[199,logseq____"^Vlogseq____",159,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[199,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[199,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[199,logseq____"^Hlogseq____",174,536870922]],[logseq____"^15logseq____",[199,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[199,logseq____"^;logseq____",logseq____"~u65297d30-d1ff-437b-869c-f50f81f214d4logseq____",536870922]],[logseq____"^15logseq____",[200,logseq____"^Qlogseq____",logseq____"C. Meinel, M. Mundhenk, [[\\logseq____"Mathematische Grundlagen der Informatik\\logseq____", 5. 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[[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[201,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[201,logseq____"^Flogseq____",195,536870922]],[logseq____"^15logseq____",[201,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[201,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[201,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[201,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[201,logseq____"^Hlogseq____",176,536870922]],[logseq____"^15logseq____",[201,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[201,logseq____"^;logseq____",logseq____"~u65297d30-1ca4-4195-8630-ce22381f7ac8logseq____",536870922]],[logseq____"^15logseq____",[202,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 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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____",536870923]],[logseq____"^15logseq____",[241,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[241,logseq____"^Flogseq____",254,536870923]],[logseq____"^15logseq____",[241,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[241,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[241,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[241,logseq____"^;logseq____",logseq____"~u65297d31-d475-4cae-b066-b4e8bda6c9a3logseq____",536870923]],[logseq____"^15logseq____",[242,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists 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falsch}\\n\\\\end{align}\\n[Beweis]([[Beweise]]):logseq____",536870923]],[logseq____"^15logseq____",[242,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[242,logseq____"^Flogseq____",245,536870923]],[logseq____"^15logseq____",[242,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[242,logseq____"^Vlogseq____",249,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",221,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",221,536870923]],[logseq____"^15logseq____",[242,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[242,logseq____"^;logseq____",logseq____"~u65297d31-c138-47e4-b29b-a1f963b3b144logseq____",536870923]],[logseq____"^15logseq____",[243,logseq____"^Qlogseq____",logseq____"#FIXMElogseq____",536870923]],[logseq____"^15logseq____",[243,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[243,logseq____"^Flogseq____",253,536870923]],[logseq____"^15logseq____",[243,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[243,logseq____"^Vlogseq____",253,536870923]],[logseq____"^15logseq____",[243,logseq____"^Ulogseq____",42,536870923]],[logseq____"^15logseq____",[243,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[243,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[243,logseq____"^Hlogseq____",42,536870923]],[logseq____"^15logseq____",[243,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[243,logseq____"^;logseq____",logseq____"~u65297d31-6f53-43d3-904e-40f606f56eb8logseq____",536870923]],[logseq____"^15logseq____",[244,logseq____"^Qlogseq____",logseq____"wobei $\\\\pi$ eine \\logseq____"[[Permutation]]\\logseq____" der Zahl $(1,2,\\\\cdots,n)$ istlogseq____",536870923]],[logseq____"^15logseq____",[244,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[244,logseq____"^Flogseq____",239,536870923]],[logseq____"^15logseq____",[244,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[244,logseq____"^Vlogseq____",239,536870923]],[logseq____"^15logseq____",[244,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[244,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[244,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[244,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[244,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[244,logseq____"^Hlogseq____",224,536870923]],[logseq____"^15logseq____",[244,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[244,logseq____"^;logseq____",logseq____"~u65297d31-1af4-48e7-b5a8-0cd9cd8f0e4elogseq____",536870923]],[logseq____"^15logseq____",[245,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist wahr}\\n\\\\end{align}\\n[Beweis]([[Beweise]]):logseq____",536870923]],[logseq____"^15logseq____",[245,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[245,logseq____"^Flogseq____",249,536870923]],[logseq____"^15logseq____",[245,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[245,logseq____"^Vlogseq____",249,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",221,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[245,logseq____"^Hlogseq____",221,536870923]],[logseq____"^15logseq____",[245,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[245,logseq____"^;logseq____",logseq____"~u65297d31-e76e-4818-bafa-bd896148a740logseq____",536870923]],[logseq____"^15logseq____",[246,logseq____"^Qlogseq____",logseq____"[Gleichartige]([[Gleichartig]]) [[Quantoren]] dürfen vertauscht werdenlogseq____",536870923]],[logseq____"^15logseq____",[246,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[246,logseq____"^Flogseq____",260,536870923]],[logseq____"^15logseq____",[246,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[246,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[246,logseq____"^Hlogseq____",218,536870923]],[logseq____"^15logseq____",[246,logseq____"^Hlogseq____",222,536870923]],[logseq____"^15logseq____",[246,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[246,logseq____"^;logseq____",logseq____"~u65297d31-5f2a-4f4c-805e-ab85270540d8logseq____",536870923]],[logseq____"^15logseq____",[247,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____",536870923]],[logseq____"^15logseq____",[247,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[247,logseq____"^Flogseq____",261,536870923]],[logseq____"^15logseq____",[247,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[247,logseq____"^Vlogseq____",217,536870923]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[247,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[247,logseq____"^;logseq____",logseq____"~u65297d31-53e5-4608-b26b-d1c9d3670820logseq____",536870923]],[logseq____"^15logseq____",[248,logseq____"^Qlogseq____",logseq____"[[Aussageform]] (AF) über [Universen]([[Universen (Logik)]]) $U_1,\\\\cdots,U_n$:logseq____",536870923]],[logseq____"^15logseq____",[248,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[248,logseq____"^Flogseq____",262,536870923]],[logseq____"^15logseq____",[248,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[248,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[248,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[248,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[248,logseq____"^Ulogseq____",232,536870923]],[logseq____"^15logseq____",[248,logseq____"^Hlogseq____",228,536870923]],[logseq____"^15logseq____",[248,logseq____"^Hlogseq____",232,536870923]],[logseq____"^15logseq____",[248,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[248,logseq____"^;logseq____",logseq____"~u65297d31-84a9-41a9-abd3-09df30c5a3f2logseq____",536870923]],[logseq____"^15logseq____",[249,logseq____"^Qlogseq____",logseq____"Betrachte $\\logseq____"xlogseq____<y\\logseq____"$ als AF über $\\\\Reals, \\\\Reals$logseq____",536870923]],[logseq____"^15logseq____",[249,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[249,logseq____"^Flogseq____",272,536870923]],[logseq____"^15logseq____",[249,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[249,logseq____"^Vlogseq____",272,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[249,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[249,logseq____"^;logseq____",logseq____"~u65297d31-3db5-42a2-8954-e7d385fbdab2logseq____",536870923]],[logseq____"^15logseq____",[250,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____",536870923]],[logseq____"^15logseq____",[250,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[250,logseq____"^Flogseq____",270,536870923]],[logseq____"^15logseq____",[250,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[250,logseq____"^Vlogseq____",270,536870923]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[250,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[250,logseq____"^;logseq____",logseq____"~u65297d31-e612-4534-bb59-298cc7afdb02logseq____",536870923]],[logseq____"^15logseq____",[251,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____",536870923]],[logseq____"^15logseq____",[251,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[251,logseq____"^Flogseq____",269,536870923]],[logseq____"^15logseq____",[251,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[251,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[251,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[251,logseq____"^;logseq____",logseq____"~u65297d31-ca20-4be2-9b2a-31e3855850a1logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Qlogseq____",logseq____"see: [[Große Azzoziativgesetze]] [[Große de Morgansche Regeln]] [[Negationsregeln]]logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Flogseq____",273,536870923]],[logseq____"^15logseq____",[252,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[252,logseq____"^Vlogseq____",273,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",213,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",225,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",230,536870923]],[logseq____"^15logseq____",[252,logseq____"^Hlogseq____",213,536870923]],[logseq____"^15logseq____",[252,logseq____"^Hlogseq____",225,536870923]],[logseq____"^15logseq____",[252,logseq____"^Hlogseq____",230,536870923]],[logseq____"^15logseq____",[252,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[252,logseq____"^;logseq____",logseq____"~u65297d31-cd85-4831-ad79-f10117b8abeclogseq____",536870923]],[logseq____"^15logseq____",[253,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____",536870923]],[logseq____"^15logseq____",[253,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[253,logseq____"^Flogseq____",256,536870923]],[logseq____"^15logseq____",[253,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[253,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[253,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[253,logseq____"^;logseq____",logseq____"~u65297d31-cd9e-4abd-a3f3-b76ce74fe10alogseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Qlogseq____",logseq____"Die beiden Quantoren $\\\\forall, \\\\exists$ erweitern die [Sprache]([[Sprachen]]) der [[Aussagenlogik]]logseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Flogseq____",259,536870923]],[logseq____"^15logseq____",[254,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[254,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",214,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",233,536870923]],[logseq____"^15logseq____",[254,logseq____"^Hlogseq____",214,536870923]],[logseq____"^15logseq____",[254,logseq____"^Hlogseq____",233,536870923]],[logseq____"^15logseq____",[254,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[254,logseq____"^;logseq____",logseq____"~u65297d31-38b5-48f2-ad66-cdaf0d79fe93logseq____",536870923]],[logseq____"^15logseq____",[255,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____",536870923]],[logseq____"^15logseq____",[255,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[255,logseq____"^Flogseq____",258,536870923]],[logseq____"^15logseq____",[255,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[255,logseq____"^Vlogseq____",258,536870923]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[255,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[255,logseq____"^;logseq____",logseq____"~u65297d31-a9f5-411d-97c1-a44d77c02728logseq____",536870923]],[logseq____"^15logseq____",[256,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____",536870923]],[logseq____"^15logseq____",[256,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[256,logseq____"^Flogseq____",251,536870923]],[logseq____"^15logseq____",[256,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[256,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",166,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[256,logseq____"^Hlogseq____",166,536870923]],[logseq____"^15logseq____",[256,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[256,logseq____"^;logseq____",logseq____"~u65297d31-3ad2-4917-b39c-5df1129986f2logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Qlogseq____",logseq____"Wäre $y$ aus $\\\\Reals$ mit $xlogseq____<y$ für alle $x$ aus $\\\\Reals$, so insbesondere auch $ylogseq____<y$, Widerspruch. $\\\\blacksquare$logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Flogseq____",242,536870923]],[logseq____"^15logseq____",[257,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[257,logseq____"^Vlogseq____",242,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",221,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[257,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[257,logseq____"^;logseq____",logseq____"~u65297d31-3583-44cb-b1cc-52bd88481a32logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Qlogseq____",logseq____"Schachtelung von quantoren ([Beispiel]([[Beispiele]])):logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Flogseq____",273,536870923]],[logseq____"^15logseq____",[258,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[258,logseq____"^Vlogseq____",260,536870923]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[258,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[258,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[258,logseq____"^;logseq____",logseq____"~u65297d31-040b-4403-96f7-ff041972de92logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Qlogseq____",logseq____"Sei $p(x)$ AF in der Variablen $x$ über $U$:logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Flogseq____",270,536870923]],[logseq____"^15logseq____",[259,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[259,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[259,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[259,logseq____"^;logseq____",logseq____"~u65297d31-2140-4bdc-92b1-793e0efea580logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Qlogseq____",logseq____"## [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Flogseq____",253,536870923]],[logseq____"^15logseq____",[260,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[260,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[260,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870923]],[logseq____"^15logseq____",[260,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[260,logseq____"^Hlogseq____",210,536870923]],[logseq____"^15logseq____",[260,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[260,logseq____"^;logseq____",logseq____"~u65297d31-fcc9-419c-842c-949a9e511e69logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Qlogseq____",logseq____"# [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Flogseq____",217,536870923]],[logseq____"^15logseq____",[261,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[261,logseq____"^Vlogseq____",217,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[261,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[261,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[261,logseq____"^Hlogseq____",210,536870923]],[logseq____"^15logseq____",[261,logseq____"^17logseq____",false,536870923]],[logseq____"^15logseq____",[261,logseq____"^;logseq____",logseq____"~u65297d31-116d-410f-94c3-07cd2e681e2flogseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Qlogseq____",logseq____"# [[Aussageform]]logseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Flogseq____",247,536870923]],[logseq____"^15logseq____",[262,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[262,logseq____"^Vlogseq____",217,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[262,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[262,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[262,logseq____"^Hlogseq____",228,536870923]],[logseq____"^15logseq____",[262,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[262,logseq____"^;logseq____",logseq____"~u65297d31-cab6-42a4-b687-aad1c207059blogseq____",536870923]],[logseq____"^15logseq____",[263,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____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Flogseq____",259,536870923]],[logseq____"^15logseq____",[263,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[263,logseq____"^Vlogseq____",259,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",166,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",235,536870923]],[logseq____"^15logseq____",[263,logseq____"^Hlogseq____",166,536870923]],[logseq____"^15logseq____",[263,logseq____"^Hlogseq____",235,536870923]],[logseq____"^15logseq____",[263,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[263,logseq____"^;logseq____",logseq____"~u65297d31-1fb8-4974-ac78-77cfb0c28c1blogseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Qlogseq____",logseq____"Verschiedene [[Quantoren]] dürfen **nicht** vertauscht werdenlogseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Flogseq____",244,536870923]],[logseq____"^15logseq____",[264,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[264,logseq____"^Vlogseq____",239,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[264,logseq____"^Hlogseq____",222,536870923]],[logseq____"^15logseq____",[264,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[264,logseq____"^;logseq____",logseq____"~u65297d31-3079-4a14-8385-56d702cd4f34logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Qlogseq____",logseq____"see: [[Allquantor]] [[Existenzquantor]]logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Flogseq____",251,536870923]],[logseq____"^15logseq____",[265,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[265,logseq____"^Vlogseq____",251,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",229,536870923]],[logseq____"^15logseq____",[265,logseq____"^Hlogseq____",223,536870923]],[logseq____"^15logseq____",[265,logseq____"^Hlogseq____",229,536870923]],[logseq____"^15logseq____",[265,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[265,logseq____"^;logseq____",logseq____"~u65297d31-fa97-42c1-beac-627b7a5afaf3logseq____",536870923]],[logseq____"^15logseq____",[266,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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Setze $y:=x+1$ (ist aus $\\\\Reals$). Dann ist $xlogseq____<x+1=y$ also $xlogseq____<y$. $\\\\blacksquare$logseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Flogseq____",245,536870923]],[logseq____"^15logseq____",[267,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[267,logseq____"^Vlogseq____",245,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",221,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[267,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[267,logseq____"^;logseq____",logseq____"~u65297d31-e00f-41c8-a6d1-0594b9942ef2logseq____",536870923]],[logseq____"^15logseq____",[268,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870923]],[logseq____"^15logseq____",[268,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[268,logseq____"^Flogseq____",246,536870923]],[logseq____"^15logseq____",[268,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[268,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[268,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[268,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[268,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[268,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[268,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[268,logseq____"^;logseq____",logseq____"~u65297d31-60c0-4024-b924-f3c59e911363logseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Qlogseq____",logseq____"Endliches Universum $U$ mit Objekten $O_1,\\\\cdots,O_m$logseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Flogseq____",241,536870923]],[logseq____"^15logseq____",[269,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[269,logseq____"^Vlogseq____",262,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[269,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[269,logseq____"^;logseq____",logseq____"~u65297d31-d211-4a13-9408-c00f53db740blogseq____",536870923]],[logseq____"^15logseq____",[270,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 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umgekehrtlogseq____",536870929]],[logseq____"^15logseq____",[306,logseq____"^Ologseq____",logseq____"^16logseq____",536870929]],[logseq____"^15logseq____",[306,logseq____"^Flogseq____",305,536870929]],[logseq____"^15logseq____",[306,logseq____"^Xlogseq____",108,536870929]],[logseq____"^15logseq____",[306,logseq____"^Vlogseq____",305,536870929]],[logseq____"^15logseq____",[306,logseq____"^Ulogseq____",108,536870929]],[logseq____"^15logseq____",[306,logseq____"^Ulogseq____",302,536870929]],[logseq____"^15logseq____",[306,logseq____"^Ulogseq____",303,536870929]],[logseq____"^15logseq____",[306,logseq____"^Ulogseq____",304,536870929]],[logseq____"^15logseq____",[306,logseq____"^Hlogseq____",302,536870929]],[logseq____"^15logseq____",[306,logseq____"^Hlogseq____",303,536870929]],[logseq____"^15logseq____",[306,logseq____"^Hlogseq____",304,536870929]],[logseq____"^15logseq____",[306,logseq____"^17logseq____",true,536870929]],[logseq____"^15logseq____",[306,logseq____"^;logseq____",logseq____"~u65297d31-e069-4c83-aa07-368075224c31logseq____",536870929]],[logseq____"^15logseq____",[307,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\logseq____"logseq____>\\logseq____"logseq____&\\\\rightarrow\\logseq____"logseq____<\\logseq____"\\\\\\\\\\n\\logseq____"logseq____<\\logseq____"logseq____&\\\\rightarrow\\logseq____"logseq____>\\logseq____"\\\\\\\\\\n\\logseq____"\\\\ge\\logseq____"logseq____&\\\\rightarrow\\logseq____"\\\\le\\logseq____"\\\\\\\\\\n\\logseq____"\\\\le\\logseq____"logseq____&\\\\rightarrow\\logseq____"\\\\ge\\logseq____"\\\\\\\\\\n\\\\end{align}logseq____",536870929]],[logseq____"^15logseq____",[307,logseq____"^Ologseq____",logseq____"^16logseq____",536870929]],[logseq____"^15logseq____",[307,logseq____"^Flogseq____",306,536870929]],[logseq____"^15logseq____",[307,logseq____"^Xlogseq____",108,536870929]],[logseq____"^15logseq____",[307,logseq____"^Vlogseq____",306,536870929]],[logseq____"^15logseq____",[307,logseq____"^Ulogseq____",108,536870929]],[logseq____"^15logseq____",[307,logseq____"^Ulogseq____",302,536870929]],[logseq____"^15logseq____",[307,logseq____"^Ulogseq____",303,536870929]],[logseq____"^15logseq____",[307,logseq____"^Ulogseq____",304,536870929]],[logseq____"^15logseq____",[307,logseq____"^17logseq____",true,536870929]],[logseq____"^15logseq____",[307,logseq____"^;logseq____",logseq____"~u65297d31-6a51-479a-802a-5a83945db69elogseq____",536870929]],[logseq____"^15logseq____",[308,logseq____"^Qlogseq____",logseq____"logseq____",536870929]],[logseq____"^15logseq____",[308,logseq____"^Ologseq____",logseq____"^16logseq____",536870929]],[logseq____"^15logseq____",[308,logseq____"^Flogseq____",307,536870929]],[logseq____"^15logseq____",[308,logseq____"^Xlogseq____",108,536870929]],[logseq____"^15logseq____",[308,logseq____"^Vlogseq____",306,536870929]],[logseq____"^15logseq____",[308,logseq____"^Ulogseq____",108,536870929]],[logseq____"^15logseq____",[308,logseq____"^Ulogseq____",302,536870929]],[logseq____"^15logseq____",[308,logseq____"^Ulogseq____",303,536870929]],[logseq____"^15logseq____",[308,logseq____"^Ulogseq____",304,536870929]],[logseq____"^15logseq____",[308,logseq____"^17logseq____",true,536870929]],[logseq____"^15logseq____",[308,logseq____"^;logseq____",logseq____"~u65297d31-c800-41ff-b63d-722ce9e2ec0clogseq____",536870929]],[logseq____"^15logseq____",[310,logseq____"^Klogseq____",1697217841300,536870930]],[logseq____"^15logseq____",[310,logseq____"^@logseq____",false,536870930]],[logseq____"^15logseq____",[310,logseq____"^Ylogseq____",logseq____"vorlesungs notizenlogseq____",536870930]],[logseq____"^15logseq____",[310,logseq____"^11logseq____",logseq____"Vorlesungs Notizenlogseq____",536870930]],[logseq____"^15logseq____",[310,logseq____"^Blogseq____",1697217841300,536870930]],[logseq____"^15logseq____",[310,logseq____"^;logseq____",logseq____"~u65297d31-e34c-4a10-a5bb-67732c6b2238logseq____",536870930]],[logseq____"^15logseq____",[311,logseq____"^Qlogseq____",logseq____"each \\logseq____".excalidraw\\logseq____" drawing file should have an equivalent \\logseq____".svg\\logseq____" file in logseq____<vault pathlogseq____>/assets/excalidraw_svg with the same basenamelogseq____",536870930]],[logseq____"^15logseq____",[311,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[311,logseq____"^Flogseq____",320,536870930]],[logseq____"^15logseq____",[311,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[311,logseq____"^Vlogseq____",318,536870930]],[logseq____"^15logseq____",[311,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[311,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[311,logseq____"^;logseq____",logseq____"~u65297d31-7b62-4fe9-a45b-ec9c22b42eb0logseq____",536870930]],[logseq____"^15logseq____",[312,logseq____"^Qlogseq____",logseq____"### Example\\n```\\nlogseq____<vault pathlogseq____>/draws/2023-10-08-21-23-31.excalidraw\\nlogseq____<vault pathlogseq____>/assets/excalidraw_svg/2023-10-08-21-23-31.svg\\n```logseq____",536870930]],[logseq____"^15logseq____",[312,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[312,logseq____"^Flogseq____",311,536870930]],[logseq____"^15logseq____",[312,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[312,logseq____"^Vlogseq____",318,536870930]],[logseq____"^15logseq____",[312,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[312,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870930]],[logseq____"^15logseq____",[312,logseq____"^Jlogseq____",[],536870930]],[logseq____"^15logseq____",[312,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[312,logseq____"^;logseq____",logseq____"~u65297d31-8613-48da-8440-a30a3b261599logseq____",536870930]],[logseq____"^15logseq____",[313,logseq____"^Qlogseq____",logseq____"Click \\logseq____"SVG\\logseq____" and copy to logseq____<vault pathlogseq____>/assets/excalidraw_svgslogseq____",536870930]],[logseq____"^15logseq____",[313,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[313,logseq____"^Flogseq____",314,536870930]],[logseq____"^15logseq____",[313,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[313,logseq____"^Vlogseq____",319,536870930]],[logseq____"^15logseq____",[313,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[313,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[313,logseq____"^;logseq____",logseq____"~u65297d31-67f4-42c9-acb8-ec9b333f5212logseq____",536870930]],[logseq____"^15logseq____",[314,logseq____"^Qlogseq____",logseq____"Enable \\logseq____"Background\\logseq____" and check if you need Dark or Light themelogseq____",536870930]],[logseq____"^15logseq____",[314,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[314,logseq____"^Flogseq____",319,536870930]],[logseq____"^15logseq____",[314,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[314,logseq____"^Vlogseq____",319,536870930]],[logseq____"^15logseq____",[314,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[314,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[314,logseq____"^;logseq____",logseq____"~u65297d31-3473-4463-800a-ba069662214alogseq____",536870930]],[logseq____"^15logseq____",[315,logseq____"^Qlogseq____",logseq____"Rename the file so the base names matchlogseq____",536870930]],[logseq____"^15logseq____",[315,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[315,logseq____"^Flogseq____",313,536870930]],[logseq____"^15logseq____",[315,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[315,logseq____"^Vlogseq____",319,536870930]],[logseq____"^15logseq____",[315,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[315,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[315,logseq____"^;logseq____",logseq____"~u65297d31-d956-49cc-b890-0af39b0e7feflogseq____",536870930]],[logseq____"^15logseq____",[316,logseq____"^Qlogseq____",logseq____"logseq____<h1 style=\\logseq____"color: red;\\logseq____"logseq____>Some drawings were not loadedlogseq____</h1logseq____>logseq____",536870930]],[logseq____"^15logseq____",[316,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[316,logseq____"^Flogseq____",321,536870930]],[logseq____"^15logseq____",[316,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[316,logseq____"^Vlogseq____",310,536870930]],[logseq____"^15logseq____",[316,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[316,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[316,logseq____"^;logseq____",logseq____"~u65297d31-0543-447b-b8e3-f76e771cc50dlogseq____",536870930]],[logseq____"^15logseq____",[317,logseq____"^Qlogseq____",logseq____"[[Auffrischung Mathematik]]logseq____",536870930]],[logseq____"^15logseq____",[317,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[317,logseq____"^Flogseq____",310,536870930]],[logseq____"^15logseq____",[317,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[317,logseq____"^Vlogseq____",310,536870930]],[logseq____"^15logseq____",[317,logseq____"^Ulogseq____",23,536870930]],[logseq____"^15logseq____",[317,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[317,logseq____"^Hlogseq____",23,536870930]],[logseq____"^15logseq____",[317,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[317,logseq____"^;logseq____",logseq____"~u65297d31-f44f-4fab-b20a-2ff438ad87a4logseq____",536870930]],[logseq____"^15logseq____",[318,logseq____"^Qlogseq____",logseq____"## How to fix as a Contributorlogseq____",536870930]],[logseq____"^15logseq____",[318,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[318,logseq____"^Flogseq____",316,536870930]],[logseq____"^15logseq____",[318,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[318,logseq____"^Vlogseq____",316,536870930]],[logseq____"^15logseq____",[318,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[318,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870930]],[logseq____"^15logseq____",[318,logseq____"^Jlogseq____",[],536870930]],[logseq____"^15logseq____",[318,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[318,logseq____"^;logseq____",logseq____"~u65297d31-c6a8-4fa8-b9d2-b093d69a37d6logseq____",536870930]],[logseq____"^15logseq____",[319,logseq____"^Qlogseq____",logseq____"to create a missing svg file, head to https://excalidraw.com/, paste the .excalidraw file, Click the Hamburger Menu logseq____> Export Imagelogseq____",536870930]],[logseq____"^15logseq____",[319,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[319,logseq____"^Flogseq____",312,536870930]],[logseq____"^15logseq____",[319,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[319,logseq____"^Vlogseq____",318,536870930]],[logseq____"^15logseq____",[319,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[319,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[319,logseq____"^;logseq____",logseq____"~u65297d31-52ab-4c94-bf7d-aeb6096953e3logseq____",536870930]],[logseq____"^15logseq____",[320,logseq____"^Qlogseq____",logseq____"drawings are located in logseq____<vault pathlogseq____>/drawslogseq____",536870930]],[logseq____"^15logseq____",[320,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[320,logseq____"^Flogseq____",318,536870930]],[logseq____"^15logseq____",[320,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[320,logseq____"^Vlogseq____",318,536870930]],[logseq____"^15logseq____",[320,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[320,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[320,logseq____"^;logseq____",logseq____"~u65297d31-8c9c-48ed-9d23-d133f20d97c4logseq____",536870930]],[logseq____"^15logseq____",[321,logseq____"^Qlogseq____",logseq____"[[Programmierung und Algorithmen]]logseq____",536870930]],[logseq____"^15logseq____",[321,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[321,logseq____"^Flogseq____",323,536870930]],[logseq____"^15logseq____",[321,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[321,logseq____"^Vlogseq____",310,536870930]],[logseq____"^15logseq____",[321,logseq____"^Ulogseq____",290,536870930]],[logseq____"^15logseq____",[321,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[321,logseq____"^Hlogseq____",290,536870930]],[logseq____"^15logseq____",[321,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[321,logseq____"^;logseq____",logseq____"~u65297d31-edc7-4498-9b82-cfea2cc250b6logseq____",536870930]],[logseq____"^15logseq____",[322,logseq____"^Qlogseq____",logseq____"[[Mathematik für Informatiker 1]]logseq____",536870930]],[logseq____"^15logseq____",[322,logseq____"^Ologseq____",logseq____"^16logseq____",536870930]],[logseq____"^15logseq____",[322,logseq____"^Flogseq____",317,536870930]],[logseq____"^15logseq____",[322,logseq____"^Xlogseq____",310,536870930]],[logseq____"^15logseq____",[322,logseq____"^Vlogseq____",310,536870930]],[logseq____"^15logseq____",[322,logseq____"^Ulogseq____",279,536870930]],[logseq____"^15logseq____",[322,logseq____"^Ulogseq____",310,536870930]],[logseq____"^15logseq____",[322,logseq____"^Hlogseq____",279,536870930]],[logseq____"^15logseq____",[322,logseq____"^17logseq____",true,536870930]],[logseq____"^15logseq____",[322,logseq____"^;logseq____",logseq____"~u65297d31-f496-4f6a-abaa-fcaefa86e323logseq____",536870930]],[logseq____"^15logseq____",[323,logseq____"^Qlogseq____",logseq____"[[Grundlagen und Diskrete 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