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logseq____&logseq____& 7 logseq____&logseq____&21 logseq____&logseq____& 35 logseq____&logseq____& 35 logseq____&logseq____& \\\\cdots\\\\end{array}logseq____",536870918]],[logseq____"^15logseq____",[33,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[33,logseq____"^Flogseq____",59,536870918]],[logseq____"^15logseq____",[33,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[33,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[33,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[33,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[33,logseq____"^;logseq____",logseq____"~u65257d4e-894c-41ec-a27c-92b8573d1463logseq____",536870918]],[logseq____"^15logseq____",[34,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____",536870918]],[logseq____"^15logseq____",[34,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[34,logseq____"^Flogseq____",64,536870918]],[logseq____"^15logseq____",[34,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[34,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[34,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[34,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[34,logseq____"^;logseq____",logseq____"~u65257d4e-03f3-4a91-979f-eb9dac022158logseq____",536870918]],[logseq____"^15logseq____",[35,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiele}$\\n\\\\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____",536870918]],[logseq____"^15logseq____",[35,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[35,logseq____"^Flogseq____",43,536870918]],[logseq____"^15logseq____",[35,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[35,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[35,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[35,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[35,logseq____"^;logseq____",logseq____"~u65257d4e-fce1-4375-8746-071349e68960logseq____",536870918]],[logseq____"^15logseq____",[36,logseq____"^Qlogseq____",logseq____"## [[Binomialkoeffizient]]logseq____",536870918]],[logseq____"^15logseq____",[36,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[36,logseq____"^Flogseq____",66,536870918]],[logseq____"^15logseq____",[36,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[36,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[36,logseq____"^Ulogseq____",26,536870918]],[logseq____"^15logseq____",[36,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[36,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"~:headinglogseq____",2],536870918]],[logseq____"^15logseq____",[36,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[36,logseq____"^Hlogseq____",26,536870918]],[logseq____"^15logseq____",[36,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[36,logseq____"^;logseq____",logseq____"~u65257d4e-6e73-42aa-bbaa-a174606f1b5clogseq____",536870918]],[logseq____"^15logseq____",[37,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Wurzeln]]logseq____",536870918]],[logseq____"^15logseq____",[37,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[37,logseq____"^Flogseq____",58,536870918]],[logseq____"^15logseq____",[37,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[37,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[37,logseq____"^Ulogseq____",29,536870918]],[logseq____"^15logseq____",[37,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[37,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870918]],[logseq____"^15logseq____",[37,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[37,logseq____"^Hlogseq____",29,536870918]],[logseq____"^15logseq____",[37,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[37,logseq____"^;logseq____",logseq____"~u65257d4e-c6ce-4cac-b161-f5c5e99df15flogseq____",536870918]],[logseq____"^15logseq____",[38,logseq____"^Qlogseq____",logseq____"#FIXME $\\\\sin^{-2}x+cos^2y=1,r$ ist falschlogseq____",536870918]],[logseq____"^15logseq____",[38,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[38,logseq____"^Flogseq____",44,536870918]],[logseq____"^15logseq____",[38,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[38,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[38,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[38,logseq____"^Ulogseq____",32,536870918]],[logseq____"^15logseq____",[38,logseq____"^Hlogseq____",32,536870918]],[logseq____"^15logseq____",[38,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[38,logseq____"^;logseq____",logseq____"~u65257d4e-7572-4d2a-a83b-9c9a6e32fef3logseq____",536870918]],[logseq____"^15logseq____",[39,logseq____"^Qlogseq____",logseq____"# [[Potenzen]]logseq____",536870918]],[logseq____"^15logseq____",[39,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[39,logseq____"^Flogseq____",46,536870918]],[logseq____"^15logseq____",[39,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[39,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[39,logseq____"^Ulogseq____",23,536870918]],[logseq____"^15logseq____",[39,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[39,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870918]],[logseq____"^15logseq____",[39,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[39,logseq____"^Hlogseq____",23,536870918]],[logseq____"^15logseq____",[39,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[39,logseq____"^;logseq____",logseq____"~u65257d4e-acda-484a-a45a-7557496d6900logseq____",536870918]],[logseq____"^15logseq____",[40,logseq____"^Qlogseq____",logseq____"# [[Binomische Formeln]]logseq____",536870918]],[logseq____"^15logseq____",[40,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[40,logseq____"^Flogseq____",38,536870918]],[logseq____"^15logseq____",[40,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[40,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[40,logseq____"^Ulogseq____",27,536870918]],[logseq____"^15logseq____",[40,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[40,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870918]],[logseq____"^15logseq____",[40,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[40,logseq____"^Hlogseq____",27,536870918]],[logseq____"^15logseq____",[40,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[40,logseq____"^;logseq____",logseq____"~u65257d4e-aa37-4339-b565-2761b7623f19logseq____",536870918]],[logseq____"^15logseq____",[41,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Logarithmen]]logseq____",536870918]],[logseq____"^15logseq____",[41,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[41,logseq____"^Flogseq____",55,536870918]],[logseq____"^15logseq____",[41,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[41,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[41,logseq____"^Ulogseq____",25,536870918]],[logseq____"^15logseq____",[41,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[41,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870918]],[logseq____"^15logseq____",[41,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[41,logseq____"^Hlogseq____",25,536870918]],[logseq____"^15logseq____",[41,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[41,logseq____"^;logseq____",logseq____"~u65257d4e-a9be-42ad-b9a1-ddf2f8689340logseq____",536870918]],[logseq____"^15logseq____",[42,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Potenzen]]logseq____",536870918]],[logseq____"^15logseq____",[42,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[42,logseq____"^Flogseq____",31,536870918]],[logseq____"^15logseq____",[42,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[42,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[42,logseq____"^Ulogseq____",23,536870918]],[logseq____"^15logseq____",[42,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[42,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870918]],[logseq____"^15logseq____",[42,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[42,logseq____"^Hlogseq____",23,536870918]],[logseq____"^15logseq____",[42,logseq____"^17logseq____",false,536870918]],[logseq____"^15logseq____",[42,logseq____"^;logseq____",logseq____"~u65257d4e-d86e-4405-aaf9-cb541b984769logseq____",536870918]],[logseq____"^15logseq____",[43,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____",536870918]],[logseq____"^15logseq____",[43,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[43,logseq____"^Flogseq____",39,536870918]],[logseq____"^15logseq____",[43,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[43,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[43,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[43,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[43,logseq____"^;logseq____",logseq____"~u65257d4e-8944-4c26-b991-456dd8a38d50logseq____",536870918]],[logseq____"^15logseq____",[44,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____",536870918]],[logseq____"^15logseq____",[44,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[44,logseq____"^Flogseq____",34,536870918]],[logseq____"^15logseq____",[44,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[44,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[44,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[44,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[44,logseq____"^;logseq____",logseq____"~u65257d4e-9545-457d-98fd-beee50eaa933logseq____",536870918]],[logseq____"^15logseq____",[45,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____",536870918]],[logseq____"^15logseq____",[45,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[45,logseq____"^Flogseq____",51,536870918]],[logseq____"^15logseq____",[45,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[45,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[45,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[45,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[45,logseq____"^;logseq____",logseq____"~u65257d4e-30df-429b-9f00-fe50c4cc01cdlogseq____",536870918]],[logseq____"^15logseq____",[46,logseq____"^Qlogseq____",logseq____"\\\\begin{array}{c}\\n{0\\\\choose 0}\\n\\\\\\\\\\n{1\\\\choose 0} {1\\\\choose 1}\\n\\\\\\\\\\n{2\\\\choose 0} {2\\\\choose 1} {2\\\\choose 2}\\n\\\\\\\\\\n{3\\\\choose 0} {3\\\\choose 1} {3\\\\choose 2} {3\\\\choose 3}\\n\\\\\\\\\\n\\\\cdots\\n\\\\end{array}logseq____",536870918]],[logseq____"^15logseq____",[46,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[46,logseq____"^Flogseq____",61,536870918]],[logseq____"^15logseq____",[46,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[46,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[46,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[46,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[46,logseq____"^;logseq____",logseq____"~u65257d4e-5e8b-4677-8a02-888406eb1d17logseq____",536870918]],[logseq____"^15logseq____",[47,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____",536870918]],[logseq____"^15logseq____",[47,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[47,logseq____"^Flogseq____",56,536870918]],[logseq____"^15logseq____",[47,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[47,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[47,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[47,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[47,logseq____"^;logseq____",logseq____"~u65257d4e-ac86-44a4-8a45-89173e5e6c0clogseq____",536870918]],[logseq____"^15logseq____",[48,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____",536870918]],[logseq____"^15logseq____",[48,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[48,logseq____"^Flogseq____",37,536870918]],[logseq____"^15logseq____",[48,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[48,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[48,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[48,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[48,logseq____"^;logseq____",logseq____"~u65257d4e-7877-4f55-b745-6b127794649alogseq____",536870918]],[logseq____"^15logseq____",[49,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____",536870918]],[logseq____"^15logseq____",[49,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[49,logseq____"^Flogseq____",41,536870918]],[logseq____"^15logseq____",[49,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[49,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[49,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[49,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[49,logseq____"^;logseq____",logseq____"~u65257d4e-924b-4f70-a830-56af5b4551d8logseq____",536870918]],[logseq____"^15logseq____",[50,logseq____"^Qlogseq____",logseq____"# [[Wurzeln]]logseq____",536870918]],[logseq____"^15logseq____",[50,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[50,logseq____"^Flogseq____",54,536870918]],[logseq____"^15logseq____",[50,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[50,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[50,logseq____"^Ulogseq____",29,536870918]],[logseq____"^15logseq____",[50,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[50,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870918]],[logseq____"^15logseq____",[50,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[50,logseq____"^Hlogseq____",29,536870918]],[logseq____"^15logseq____",[50,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[50,logseq____"^;logseq____",logseq____"~u65257d4e-8177-43f7-aa53-39b73c350be7logseq____",536870918]],[logseq____"^15logseq____",[51,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn ergibtlogseq____",536870918]],[logseq____"^15logseq____",[51,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[51,logseq____"^Flogseq____",68,536870918]],[logseq____"^15logseq____",[51,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[51,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[51,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[51,logseq____"^Ulogseq____",32,536870918]],[logseq____"^15logseq____",[51,logseq____"^Hlogseq____",32,536870918]],[logseq____"^15logseq____",[51,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[51,logseq____"^;logseq____",logseq____"~u65257d4e-3fd4-4d88-8e7b-c307fe033798logseq____",536870918]],[logseq____"^15logseq____",[52,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____",536870918]],[logseq____"^15logseq____",[52,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[52,logseq____"^Flogseq____",35,536870918]],[logseq____"^15logseq____",[52,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[52,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[52,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[52,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[52,logseq____"^;logseq____",logseq____"~u65257d4e-2a40-4b18-8296-29ef338e2327logseq____",536870918]],[logseq____"^15logseq____",[53,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____",536870918]],[logseq____"^15logseq____",[53,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[53,logseq____"^Flogseq____",67,536870918]],[logseq____"^15logseq____",[53,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[53,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[53,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[53,logseq____"^;logseq____",logseq____"~u65257d4e-6e39-4b31-a0e3-80b5dcb61506logseq____",536870918]],[logseq____"^15logseq____",[54,logseq____"^Qlogseq____",logseq____"cosa -logseq____> beliebige Variable\\ncubus -logseq____> hoch 3logseq____",536870918]],[logseq____"^15logseq____",[54,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[54,logseq____"^Flogseq____",52,536870918]],[logseq____"^15logseq____",[54,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[54,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[54,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[54,logseq____"^;logseq____",logseq____"~u65257d4e-7423-4195-a99b-f60d2f0213b3logseq____",536870918]],[logseq____"^15logseq____",[55,logseq____"^Qlogseq____",logseq____"logseq____",536870918]],[logseq____"^15logseq____",[55,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[55,logseq____"^Flogseq____",57,536870918]],[logseq____"^15logseq____",[55,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[55,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[55,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[55,logseq____"^;logseq____",logseq____"~u65257d4e-ceae-4adb-92b6-ffece29de352logseq____",536870918]],[logseq____"^15logseq____",[56,logseq____"^Qlogseq____",logseq____"# Was ist Größer?logseq____",536870918]],[logseq____"^15logseq____",[56,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[56,logseq____"^Flogseq____",60,536870918]],[logseq____"^15logseq____",[56,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[56,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[56,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870918]],[logseq____"^15logseq____",[56,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[56,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[56,logseq____"^;logseq____",logseq____"~u65257d4e-8749-4efd-a90c-c9f4e0a00822logseq____",536870918]],[logseq____"^15logseq____",[57,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche Gleichungen sind richtig?}$\\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____",536870918]],[logseq____"^15logseq____",[57,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[57,logseq____"^Flogseq____",42,536870918]],[logseq____"^15logseq____",[57,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[57,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[57,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[57,logseq____"^;logseq____",logseq____"~u65257d4e-9958-423d-856b-62ee644c15c1logseq____",536870918]],[logseq____"^15logseq____",[58,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____",536870918]],[logseq____"^15logseq____",[58,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[58,logseq____"^Flogseq____",49,536870918]],[logseq____"^15logseq____",[58,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[58,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[58,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[58,logseq____"^;logseq____",logseq____"~u65257d4e-a616-40ce-8a9c-07838a396610logseq____",536870918]],[logseq____"^15logseq____",[59,logseq____"^Qlogseq____",logseq____"## [[Pascalsches Dreieck]]logseq____",536870918]],[logseq____"^15logseq____",[59,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[59,logseq____"^Flogseq____",53,536870918]],[logseq____"^15logseq____",[59,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[59,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",24,536870918]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[59,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870918]],[logseq____"^15logseq____",[59,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[59,logseq____"^Hlogseq____",24,536870918]],[logseq____"^15logseq____",[59,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[59,logseq____"^;logseq____",logseq____"~u65257d4e-80be-4ab1-8776-e97a22cb18d0logseq____",536870918]],[logseq____"^15logseq____",[60,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____",536870918]],[logseq____"^15logseq____",[60,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[60,logseq____"^Flogseq____",48,536870918]],[logseq____"^15logseq____",[60,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[60,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[60,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[60,logseq____"^;logseq____",logseq____"~u65257d4e-7873-462a-8cc2-ba948be6008flogseq____",536870918]],[logseq____"^15logseq____",[61,logseq____"^Qlogseq____",logseq____"Das [[Pascalsche Dreieck]] lässt sich auch mit den entsprechenden Binomialkoeffizienten darstellenlogseq____",536870918]],[logseq____"^15logseq____",[61,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[61,logseq____"^Flogseq____",36,536870918]],[logseq____"^15logseq____",[61,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[61,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",30,536870918]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[61,logseq____"^Hlogseq____",30,536870918]],[logseq____"^15logseq____",[61,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[61,logseq____"^;logseq____",logseq____"~u65257d4e-8ea7-4fcf-921b-cf54cee01fe9logseq____",536870918]],[logseq____"^15logseq____",[62,logseq____"^Qlogseq____",logseq____"$\\\\text{Die Lösung von } x^n-a=0 \\\\text{ ist } x=\\\\sqrt[n]a$ #FIXME ich glaube hier fehlt logseq____>0 constraintlogseq____",536870918]],[logseq____"^15logseq____",[62,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[62,logseq____"^Flogseq____",50,536870918]],[logseq____"^15logseq____",[62,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[62,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",32,536870918]],[logseq____"^15logseq____",[62,logseq____"^Hlogseq____",32,536870918]],[logseq____"^15logseq____",[62,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[62,logseq____"^;logseq____",logseq____"~u65257d4e-8ccb-41a9-9df2-92d460b35102logseq____",536870918]],[logseq____"^15logseq____",[63,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____",536870918]],[logseq____"^15logseq____",[63,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[63,logseq____"^Flogseq____",62,536870918]],[logseq____"^15logseq____",[63,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[63,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[63,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[63,logseq____"^;logseq____",logseq____"~u65257d4e-b5be-472e-8bf4-a0415bde3152logseq____",536870918]],[logseq____"^15logseq____",[64,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Trigonometrie]]logseq____",536870918]],[logseq____"^15logseq____",[64,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[64,logseq____"^Flogseq____",47,536870918]],[logseq____"^15logseq____",[64,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[64,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",28,536870918]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[64,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870918]],[logseq____"^15logseq____",[64,logseq____"^Jlogseq____",[],536870918]],[logseq____"^15logseq____",[64,logseq____"^Hlogseq____",28,536870918]],[logseq____"^15logseq____",[64,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[64,logseq____"^;logseq____",logseq____"~u65257d4e-8700-46a1-bb3f-08556f6c7517logseq____",536870918]],[logseq____"^15logseq____",[65,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiele}$\\n\\\\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____",536870918]],[logseq____"^15logseq____",[65,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[65,logseq____"^Flogseq____",33,536870918]],[logseq____"^15logseq____",[65,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[65,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[65,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[65,logseq____"^;logseq____",logseq____"~u65257d4e-e130-48b2-b775-e6acb1b58e87logseq____",536870918]],[logseq____"^15logseq____",[66,logseq____"^Qlogseq____",logseq____"#FIXME $(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5=2+20x+40x^2$ ist Falschlogseq____",536870918]],[logseq____"^15logseq____",[66,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[66,logseq____"^Flogseq____",65,536870918]],[logseq____"^15logseq____",[66,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[66,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",32,536870918]],[logseq____"^15logseq____",[66,logseq____"^Hlogseq____",32,536870918]],[logseq____"^15logseq____",[66,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[66,logseq____"^;logseq____",logseq____"~u65257d4e-a2c1-4a9c-83c3-4cc7f117a79elogseq____",536870918]],[logseq____"^15logseq____",[67,logseq____"^Qlogseq____",logseq____"$\\\\text{Allgemein}$\\n\\\\begin{align}\\n(a+b)^2logseq____&=a^2+2ab+b^2logseq____&,\\\\text{ 1. Binomische Formel}\\\\\\\\\\n(a-b)^2logseq____&=a^2-2ab+b^2logseq____&,\\\\text{ 2. Binomische Formel}\\\\\\\\\\n(a+b)(a-b)logseq____&=a^2-b^2logseq____&,\\\\text{ 3. Binomische Formel}\\\\\\\\\\n\\\\end{align}logseq____",536870918]],[logseq____"^15logseq____",[67,logseq____"^Ologseq____",logseq____"^16logseq____",536870918]],[logseq____"^15logseq____",[67,logseq____"^Flogseq____",40,536870918]],[logseq____"^15logseq____",[67,logseq____"^Xlogseq____",31,536870918]],[logseq____"^15logseq____",[67,logseq____"^Vlogseq____",31,536870918]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",31,536870918]],[logseq____"^15logseq____",[67,logseq____"^17logseq____",true,536870918]],[logseq____"^15logseq____",[67,logseq____"^;logseq____",logseq____"~u65257d4e-7e15-4f54-b8f9-cd06f3a6d96clogseq____",536870918]],[logseq____"^15logseq____",[68,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiele}$\\n\\\\begin{align}\\n\\\\sqrt{x^4+6x^2y+9y^2}logseq____&=\\\\sqrt{(x^2+3y)^2}=x^2+3y\\\\\\\\\\n\\\\sqrt {x + \\\\sqrt{x^2-y^2}}\\\\sqrt{x-\\\\sqrt{x^2-y^2}}logseq____&=\\\\sqrt{\\\\left(x + 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#FIXMElogseq____",536870920]],[logseq____"^15logseq____",[97,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[97,logseq____"^Flogseq____",98,536870920]],[logseq____"^15logseq____",[97,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[97,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",32,536870920]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",90,536870920]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",91,536870920]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[97,logseq____"^Hlogseq____",32,536870920]],[logseq____"^15logseq____",[97,logseq____"^Hlogseq____",90,536870920]],[logseq____"^15logseq____",[97,logseq____"^Hlogseq____",91,536870920]],[logseq____"^15logseq____",[97,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[97,logseq____"^;logseq____",logseq____"~u65257d4f-f19c-4017-a8b7-3a941a8333adlogseq____",536870920]],[logseq____"^15logseq____",[98,logseq____"^Qlogseq____",logseq____"[[Aussagenlogische variable]]: [[Variable]], die den Wert $w$ oder $f$ annimmtlogseq____",536870920]],[logseq____"^15logseq____",[98,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[98,logseq____"^Flogseq____",110,536870920]],[logseq____"^15logseq____",[98,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[98,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",78,536870920]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",87,536870920]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[98,logseq____"^Hlogseq____",78,536870920]],[logseq____"^15logseq____",[98,logseq____"^Hlogseq____",87,536870920]],[logseq____"^15logseq____",[98,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[98,logseq____"^;logseq____",logseq____"~u65257d4f-e45f-401a-a1f9-186b8f041769logseq____",536870920]],[logseq____"^15logseq____",[99,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiele}$\\n\\\\begin{align}\\np\\\\land q\\\\lor r\\\\\\\\\\n(p\\\\land q)\\\\lor r\\\\\\\\\\np\\\\land (q\\\\lor r)\\\\\\\\\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[99,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[99,logseq____"^Flogseq____",101,536870920]],[logseq____"^15logseq____",[99,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[99,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[99,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[99,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[99,logseq____"^;logseq____",logseq____"~u65257d4f-1db7-4ea9-aa0c-2f3ec7995808logseq____",536870920]],[logseq____"^15logseq____",[100,logseq____"^Qlogseq____",logseq____"logseq____",536870920]],[logseq____"^15logseq____",[100,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[100,logseq____"^Flogseq____",117,536870920]],[logseq____"^15logseq____",[100,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[100,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[100,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[100,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[100,logseq____"^;logseq____",logseq____"~u65257d4e-7271-48cc-a698-65dfd82ea870logseq____",536870920]],[logseq____"^15logseq____",[101,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____",536870920]],[logseq____"^15logseq____",[101,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[101,logseq____"^Flogseq____",112,536870920]],[logseq____"^15logseq____",[101,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[101,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[101,logseq____"^Ulogseq____",89,536870920]],[logseq____"^15logseq____",[101,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[101,logseq____"^Hlogseq____",89,536870920]],[logseq____"^15logseq____",[101,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[101,logseq____"^;logseq____",logseq____"~u65257d4f-d6db-47e7-922d-782397c79fa8logseq____",536870920]],[logseq____"^15logseq____",[102,logseq____"^Qlogseq____",logseq____"Alternativ: $p\\\\equiv q$, falls $p\\\\leftrightarrow q$ [[Tantologie]]logseq____",536870920]],[logseq____"^15logseq____",[102,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[102,logseq____"^Flogseq____",103,536870920]],[logseq____"^15logseq____",[102,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[102,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[102,logseq____"^Ulogseq____",83,536870920]],[logseq____"^15logseq____",[102,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[102,logseq____"^Hlogseq____",83,536870920]],[logseq____"^15logseq____",[102,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[102,logseq____"^;logseq____",logseq____"~u65257d4e-a44f-4355-9d27-64ca76014ccclogseq____",536870920]],[logseq____"^15logseq____",[103,logseq____"^Qlogseq____",logseq____"Formeln $p,q$ [[logisch äquivalent]], wenn sie den gleichen [[Wahrheitswerteverlauf]] haben.\\n[[Schreibweise]]: $p\\\\equiv q$logseq____",536870920]],[logseq____"^15logseq____",[103,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[103,logseq____"^Flogseq____",114,536870920]],[logseq____"^15logseq____",[103,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[103,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[103,logseq____"^Ulogseq____",77,536870920]],[logseq____"^15logseq____",[103,logseq____"^Ulogseq____",86,536870920]],[logseq____"^15logseq____",[103,logseq____"^Ulogseq____",94,536870920]],[logseq____"^15logseq____",[103,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[103,logseq____"^Hlogseq____",77,536870920]],[logseq____"^15logseq____",[103,logseq____"^Hlogseq____",86,536870920]],[logseq____"^15logseq____",[103,logseq____"^Hlogseq____",94,536870920]],[logseq____"^15logseq____",[103,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[103,logseq____"^;logseq____",logseq____"~u65257d4f-bc90-4bc6-9d30-b88b4500ba68logseq____",536870920]],[logseq____"^15logseq____",[104,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiel: Bestimmung des }$[[Wahrheitswerteverlauf]]$\\\\text{ von }(p\\\\rightarrow q)\\\\land(q\\\\rightarrow p)$\\n|$p$|$q$||$p\\\\rightarrow q$|$q\\\\rightarrow p$|$(p\\\\rightarrow q)\\\\land(q\\\\rightarrow p)|$p\\\\leftrightarrow q$|\\n|f|f||w|w|w|w|\\n|f|w||w|f|f|f|\\n|w|f||f|w|f|f|\\n|w|w||w|w|w|w|logseq____",536870920]],[logseq____"^15logseq____",[104,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[104,logseq____"^Flogseq____",115,536870920]],[logseq____"^15logseq____",[104,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[104,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[104,logseq____"^Ulogseq____",77,536870920]],[logseq____"^15logseq____",[104,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[104,logseq____"^Hlogseq____",77,536870920]],[logseq____"^15logseq____",[104,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[104,logseq____"^;logseq____",logseq____"~u65257d4f-b2ab-4582-969f-6c5bc8d27ef9logseq____",536870920]],[logseq____"^15logseq____",[105,logseq____"^Qlogseq____",logseq____"# Büchertiplogseq____",536870920]],[logseq____"^15logseq____",[105,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[105,logseq____"^Flogseq____",96,536870920]],[logseq____"^15logseq____",[105,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[105,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[105,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[105,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870920]],[logseq____"^15logseq____",[105,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[105,logseq____"^17logseq____",false,536870920]],[logseq____"^15logseq____",[105,logseq____"^;logseq____",logseq____"~u65257d4f-b0ce-4b48-93b5-4ee337770069logseq____",536870920]],[logseq____"^15logseq____",[106,logseq____"^Qlogseq____",logseq____"[[Belegung]] (der Variablen): Zuordnung von $w/f$ an jede [[Variable]] einer [[Aussageformel]]logseq____",536870920]],[logseq____"^15logseq____",[106,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[106,logseq____"^Flogseq____",97,536870920]],[logseq____"^15logseq____",[106,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[106,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[106,logseq____"^Ulogseq____",87,536870920]],[logseq____"^15logseq____",[106,logseq____"^Ulogseq____",91,536870920]],[logseq____"^15logseq____",[106,logseq____"^Ulogseq____",95,536870920]],[logseq____"^15logseq____",[106,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[106,logseq____"^Hlogseq____",87,536870920]],[logseq____"^15logseq____",[106,logseq____"^Hlogseq____",91,536870920]],[logseq____"^15logseq____",[106,logseq____"^Hlogseq____",95,536870920]],[logseq____"^15logseq____",[106,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[106,logseq____"^;logseq____",logseq____"~u65257d4f-a628-4151-b40d-76ff950b33c8logseq____",536870920]],[logseq____"^15logseq____",[107,logseq____"^Qlogseq____",logseq____"Satz, der wahr oder falsch ist, d.h. der [[Wahrheitswert]] $w$ bzw $f$ hat ($t/f$, $1/0$, $\\\\text{wahr}/\\\\text{falsch}$)logseq____",536870920]],[logseq____"^15logseq____",[107,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[107,logseq____"^Flogseq____",109,536870920]],[logseq____"^15logseq____",[107,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[107,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[107,logseq____"^Ulogseq____",81,536870920]],[logseq____"^15logseq____",[107,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[107,logseq____"^Hlogseq____",81,536870920]],[logseq____"^15logseq____",[107,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[107,logseq____"^;logseq____",logseq____"~u65257d4f-2e36-4ea0-8564-0d92aef9139clogseq____",536870920]],[logseq____"^15logseq____",[108,logseq____"^Qlogseq____",logseq____"C. Meinel, M. Mundhenk, [[\\logseq____"Mathematische Grundlagen der Informatik\\logseq____", 5. Auflage 2011]]logseq____",536870920]],[logseq____"^15logseq____",[108,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[108,logseq____"^Flogseq____",105,536870920]],[logseq____"^15logseq____",[108,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[108,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[108,logseq____"^Ulogseq____",82,536870920]],[logseq____"^15logseq____",[108,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[108,logseq____"^Hlogseq____",82,536870920]],[logseq____"^15logseq____",[108,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[108,logseq____"^;logseq____",logseq____"~u65257d4f-c73f-4395-8dcb-0b14c80e2490logseq____",536870920]],[logseq____"^15logseq____",[109,logseq____"^Qlogseq____",logseq____"# 1. [[Aussagen]]logseq____",536870920]],[logseq____"^15logseq____",[109,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[109,logseq____"^Flogseq____",116,536870920]],[logseq____"^15logseq____",[109,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[109,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[109,logseq____"^Ulogseq____",89,536870920]],[logseq____"^15logseq____",[109,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[109,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870920]],[logseq____"^15logseq____",[109,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[109,logseq____"^Hlogseq____",89,536870920]],[logseq____"^15logseq____",[109,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[109,logseq____"^;logseq____",logseq____"~u65257d4f-0c92-4046-8209-4eb4ebebb0c9logseq____",536870920]],[logseq____"^15logseq____",[110,logseq____"^Qlogseq____",logseq____"Soweit die \\logseq____"[[Syntax]]\\logseq____", jetzt die [[Semantik]]. Die [[Wahrheitswert]]e\\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____",536870920]],[logseq____"^15logseq____",[110,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[110,logseq____"^Flogseq____",99,536870920]],[logseq____"^15logseq____",[110,logseq____"^Xlogseq____",96,536870920]],[logseq____"^15logseq____",[110,logseq____"^Vlogseq____",96,536870920]],[logseq____"^15logseq____",[110,logseq____"^Ulogseq____",81,536870920]],[logseq____"^15logseq____",[110,logseq____"^Ulogseq____",85,536870920]],[logseq____"^15logseq____",[110,logseq____"^Ulogseq____",88,536870920]],[logseq____"^15logseq____",[110,logseq____"^Ulogseq____",92,536870920]],[logseq____"^15logseq____",[110,logseq____"^Ulogseq____",96,536870920]],[logseq____"^15logseq____",[110,logseq____"^Hlogseq____",81,536870920]],[logseq____"^15logseq____",[110,logseq____"^Hlogseq____",85,536870920]],[logseq____"^15logseq____",[110,logseq____"^Hlogseq____",88,536870920]],[logseq____"^15logseq____",[110,logseq____"^Hlogseq____",92,536870920]],[logseq____"^15logseq____",[110,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[110,logseq____"^;logseq____",logseq____"~u65257d4f-d62e-4815-b8fc-efa1354d0a08logseq____",536870920]],[logseq____"^15logseq____",[111,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiele}$\\n\\\\begin{align}\\n\\logseq____"5\\\\text{ ist prim}\\logseq____" logseq____&\\\\quadlogseq____& (w)\\\\\\\\\\n\\logseq____"4\\\\text{ ist prim}\\logseq____" logseq____&logseq____& (f)\\\\\\\\\\n\\logseq____"\\\\text{Jede gerade Zahl }\\\\ge4\\\\notag\\\\\\\\\\n\\\\text{ ist Summe zweier Primzahlen}\\logseq____" logseq____&logseq____& (\\\\text{Aussage, }w\\\\text{ oder }f)\\\\\\\\\\n(\\\\text{Vermutung von Goldback 1742})\\\\text{, }logseq____&logseq____&\\\\text{richtig für gerade Zahlen bis }4*10^{18}\\\\notag\\\\\\\\\\n\\logseq____"\\\\text{Dieser Satz ist falsch}\\logseq____" logseq____&logseq____& (\\\\text{keine Aussage})\\\\\\\\\\n\\logseq____"\\\\text{Die Gleichung } x^2+y^2=z^2 \\\\notag\\\\\\\\\\n\\\\text{ hat eine Lösung }x,y,z\\\\notag\\n\\\\\\\\\\\\text{ aus positiven ganzen Zahlen}\\logseq____" logseq____&logseq____& (w\\\\text{, z.B. }x=3,y=4,z=5)\\\\\\\\\\n\\logseq____"\\\\text{Für }n\\\\ge3\\\\text{ hat die Gleichung }\\\\notag\\\\\\\\\\nx^n+y^n=z^n\\\\text{ eine Lösung}logseq____&logseq____& (f\\\\text{, wie von Fermat 1640 vermutet}\\\\notag\\\\\\\\\\nx,y,z \\\\text{aus positiven ganzen Zahlen}\\logseq____" logseq____&logseq____&\\\\text{und von Wiles 1994 bewiesen})\\\\\\\\\\n\\logseq____"\\\\text{Gott ist tot}\\logseq____" logseq____&logseq____& (\\\\text{Aussage? Wohl kaum?})\\\\\\\\\\n\\logseq____"\\\\text{Nietzsche ist tot}\\logseq____" logseq____&logseq____& (\\\\text{Aussage?}) \\\\\\\\\\n\\logseq____"\\\\text{Die Länder einer Landkarte lassen sich}\\\\notag\\\\\\\\\\n\\\\text{so mit nur vier Farben färben,}\\\\notag\\\\\\\\\\n\\\\text{dass Länder mit einer gemeinsamen}\\\\notag\\\\\\\\\\n\\\\text{Grenzlienie verschieden gefärbt sind.}\\logseq____" logseq____&logseq____& (\\\\text{Aussagge; }w\\\\text{, sog 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