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62 lines
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x=-\\\\frac{11}{12}\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[53,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[53,logseq____"^Flogseq____",88,536870919]],[logseq____"^15logseq____",[53,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[53,logseq____"^Vlogseq____",88,536870919]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[53,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[53,logseq____"^;logseq____",logseq____"~u652978d9-42d3-4744-af2c-cf15dc619ee2logseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Qlogseq____",logseq____"# [[Quadratische Gleichungen]]logseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Flogseq____",98,536870919]],[logseq____"^15logseq____",[54,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^Vlogseq____",77,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[54,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[54,logseq____"^Hlogseq____",29,536870919]],[logseq____"^15logseq____",[54,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[54,logseq____"^;logseq____",logseq____"~u652978d9-ed1b-46bf-862c-77a4e81c4af1logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Qlogseq____",logseq____"# [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Flogseq____",77,536870919]],[logseq____"^15logseq____",[55,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^Vlogseq____",77,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[55,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[55,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[55,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[55,logseq____"^;logseq____",logseq____"~u652978d9-2d18-4ee3-9504-227d75339efdlogseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Qlogseq____",logseq____"Das [[Pascalsche Dreieck]] lässt sich auch mit den entsprechenden Binomialkoeffizienten darstellenlogseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Flogseq____",47,536870919]],[logseq____"^15logseq____",[56,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Vlogseq____",47,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",38,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Hlogseq____",38,536870919]],[logseq____"^15logseq____",[56,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[56,logseq____"^;logseq____",logseq____"~u652978d9-2757-49d7-9548-f12552e5f97alogseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Flogseq____",48,536870919]],[logseq____"^15logseq____",[57,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Vlogseq____",75,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[57,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[57,logseq____"^;logseq____",logseq____"~u652978d9-8144-48a0-8112-588a4c863a34logseq____",536870919]],[logseq____"^15logseq____",[58,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____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Flogseq____",95,536870919]],[logseq____"^15logseq____",[58,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^Vlogseq____",95,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[58,logseq____"^;logseq____",logseq____"~u652978d9-94f7-4e81-be18-cdc9d9d84374logseq____",536870919]],[logseq____"^15logseq____",[59,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____",[59,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Flogseq____",62,536870919]],[logseq____"^15logseq____",[59,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Vlogseq____",62,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[59,logseq____"^;logseq____",logseq____"~u652978d9-e4f4-4bd1-8fef-d18a7ded56delogseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Qlogseq____",logseq____"## [[pq-Formel]]logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Flogseq____",61,536870919]],[logseq____"^15logseq____",[60,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[60,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[60,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[60,logseq____"^Hlogseq____",43,536870919]],[logseq____"^15logseq____",[60,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[60,logseq____"^;logseq____",logseq____"~u652978d9-31ac-4451-9e91-4c9619b4db0elogseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nax^2+bx+clogseq____&=0logseq____&|:a\\\\quadlogseq____&a\\\\ne0\\\\\\\\\\nx^2+\\\\frac ba x + \\\\frac ca logseq____&= 0\\\\\\\\\\nplogseq____&=\\\\frac ba,logseq____&q = \\\\frac ca\\\\\\\\\\nx^2+px+qlogseq____&=0\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Flogseq____",54,536870919]],[logseq____"^15logseq____",[61,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[61,logseq____"^;logseq____",logseq____"~u652978d9-75ea-450f-8fd5-9de52750879elogseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Trigonometrie]]logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Flogseq____",66,536870919]],[logseq____"^15logseq____",[62,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[62,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[62,logseq____"^Hlogseq____",36,536870919]],[logseq____"^15logseq____",[62,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[62,logseq____"^;logseq____",logseq____"~u652978d9-a48f-4501-81a0-d9fc1d51d969logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Qlogseq____",logseq____"$\\\\text{Wenn }alogseq____>0\\\\text{, dann}$\\n\\\\begin{align}\\n\\\\frac b{\\\\sqrt a} = \\\\frac b{\\\\sqrt a}*\\\\frac{\\\\sqrt a}{\\\\sqrt a} = \\\\frac {b\\\\sqrt a}a\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Flogseq____",86,536870919]],[logseq____"^15logseq____",[63,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[63,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[63,logseq____"^;logseq____",logseq____"~u652978d9-a393-4ed1-85a7-3766fcf81993logseq____",536870919]],[logseq____"^15logseq____",[64,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____",[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____",96,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[64,logseq____"^;logseq____",logseq____"~u652978d9-aefa-4e61-a387-597ff6bcdc62logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Qlogseq____",logseq____"[[Satz von Vieta]]\\n\\\\begin{align}\\nx^2+px+qlogseq____&=0 logseq____&logseq____& \\\\text{habe }x_{1,2}\\\\text{ als Lösung}\\\\\\\\\\nx_1+x_2logseq____&=-p logseq____&logseq____& x_1*x_2=q\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Flogseq____",90,536870919]],[logseq____"^15logseq____",[65,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Vlogseq____",60,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[65,logseq____"^Hlogseq____",41,536870919]],[logseq____"^15logseq____",[65,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[65,logseq____"^;logseq____",logseq____"~u652978d9-03a3-4ac8-a5b3-367570b72fealogseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Qlogseq____",logseq____"# Was ist Größer?logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Flogseq____",96,536870919]],[logseq____"^15logseq____",[66,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^Vlogseq____",40,536870919]],[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____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[66,logseq____"^;logseq____",logseq____"~u652978d9-fee0-4c72-93ec-99757ce008eelogseq____",536870919]],[logseq____"^15logseq____",[67,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____",[67,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Flogseq____",74,536870919]],[logseq____"^15logseq____",[67,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Vlogseq____",74,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[67,logseq____"^;logseq____",logseq____"~u652978d9-6520-40d3-9dca-d7411cd7746clogseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Flogseq____",99,536870919]],[logseq____"^15logseq____",[68,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Vlogseq____",98,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[68,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[68,logseq____"^;logseq____",logseq____"~u652978d9-eafe-4852-9b26-f6ba73b32448logseq____",536870919]],[logseq____"^15logseq____",[69,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____",[69,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Flogseq____",55,536870919]],[logseq____"^15logseq____",[69,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^Vlogseq____",55,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[69,logseq____"^;logseq____",logseq____"~u652978d9-3b3f-4884-88e5-1291b66f9904logseq____",536870919]],[logseq____"^15logseq____",[70,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____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Flogseq____",98,536870919]],[logseq____"^15logseq____",[70,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^Vlogseq____",98,536870919]],[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____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[70,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[70,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[70,logseq____"^;logseq____",logseq____"~u652978d9-0df1-4feb-a12e-2ae4e062cde3logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Qlogseq____",logseq____"#FIXME $\\\\sin^{-2}x+cos^2y=1,r$ ist falschlogseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Flogseq____",50,536870919]],[logseq____"^15logseq____",[71,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Vlogseq____",50,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[71,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[71,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[71,logseq____"^;logseq____",logseq____"~u652978d9-2332-48f6-8f55-b843f05eaab2logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Logarithmen]]logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Flogseq____",103,536870919]],[logseq____"^15logseq____",[72,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[72,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[72,logseq____"^Hlogseq____",31,536870919]],[logseq____"^15logseq____",[72,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[72,logseq____"^;logseq____",logseq____"~u652978d9-e00b-4f07-ab8d-a96604807db4logseq____",536870919]],[logseq____"^15logseq____",[73,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____",[73,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Flogseq____",57,536870919]],[logseq____"^15logseq____",[73,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^Vlogseq____",57,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[73,logseq____"^;logseq____",logseq____"~u652978d9-b3c1-40c1-8bb4-32dfc3e5fc5flogseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Flogseq____",69,536870919]],[logseq____"^15logseq____",[74,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^Vlogseq____",55,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[74,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[74,logseq____"^;logseq____",logseq____"~u652978d9-5785-4a0e-9dd8-e298dbbb64a5logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Qlogseq____",logseq____"## [[Pascalsches Dreieck]]logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Flogseq____",83,536870919]],[logseq____"^15logseq____",[75,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^Vlogseq____",95,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[75,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[75,logseq____"^Hlogseq____",28,536870919]],[logseq____"^15logseq____",[75,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[75,logseq____"^;logseq____",logseq____"~u652978d9-bbbc-46e6-b365-da6e03ecb2e9logseq____",536870919]],[logseq____"^15logseq____",[76,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____",[76,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Flogseq____",96,536870919]],[logseq____"^15logseq____",[76,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Vlogseq____",96,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[76,logseq____"^;logseq____",logseq____"~u652978d9-d96a-48b8-9b49-2034689fa87flogseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Qlogseq____",logseq____"# [[Potenzen]] und [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Flogseq____",95,536870919]],[logseq____"^15logseq____",[77,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[77,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[77,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[77,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[77,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[77,logseq____"^;logseq____",logseq____"~u652978d9-1156-4a78-a64a-39f9b1915950logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Qlogseq____",logseq____"https://www.wolframalpha.comlogseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Flogseq____",100,536870919]],[logseq____"^15logseq____",[78,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Vlogseq____",100,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[78,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[78,logseq____"^;logseq____",logseq____"~u652978d9-a384-40f3-917c-a4e150247c7alogseq____",536870919]],[logseq____"^15logseq____",[79,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____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Flogseq____",56,536870919]],[logseq____"^15logseq____",[79,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Vlogseq____",47,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[79,logseq____"^;logseq____",logseq____"~u652978d9-1d9e-42c5-b283-f28ce7c2e001logseq____",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____",103,536870919]],[logseq____"^15logseq____",[80,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^Vlogseq____",103,536870919]],[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____"~u652978d9-a08f-430f-a076-9d83c76f5d8elogseq____",536870919]],[logseq____"^15logseq____",[81,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____",[81,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Flogseq____",94,536870919]],[logseq____"^15logseq____",[81,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^Vlogseq____",72,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[81,logseq____"^;logseq____",logseq____"~u652978d9-b55a-443e-9bdc-6301178b7b14logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn ergibtlogseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Flogseq____",51,536870919]],[logseq____"^15logseq____",[82,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^Vlogseq____",68,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[82,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[82,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[82,logseq____"^;logseq____",logseq____"~u652978d9-d8c2-41a1-9401-f776c20c2053logseq____",536870919]],[logseq____"^15logseq____",[83,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____",[83,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Flogseq____",58,536870919]],[logseq____"^15logseq____",[83,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^Vlogseq____",95,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[83,logseq____"^;logseq____",logseq____"~u652978d9-9e2c-4f2c-ab12-5670b2066646logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Flogseq____",65,536870919]],[logseq____"^15logseq____",[84,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^Vlogseq____",65,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[84,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[84,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[84,logseq____"^;logseq____",logseq____"~u652978d9-af09-4c87-b704-e11f65745581logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Flogseq____",63,536870919]],[logseq____"^15logseq____",[85,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[85,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[85,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[85,logseq____"^;logseq____",logseq____"~u652978d9-ac9e-419a-9442-daed89507bfflogseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Qlogseq____",logseq____"## [[Rationalmachen]] des [[Nenner]]slogseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Flogseq____",87,536870919]],[logseq____"^15logseq____",[86,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Vlogseq____",98,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[86,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[86,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[86,logseq____"^Hlogseq____",39,536870919]],[logseq____"^15logseq____",[86,logseq____"^Hlogseq____",45,536870919]],[logseq____"^15logseq____",[86,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[86,logseq____"^;logseq____",logseq____"~u652978d9-8bb9-4d72-8919-3614041f21b0logseq____",536870919]],[logseq____"^15logseq____",[87,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____",[87,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Flogseq____",68,536870919]],[logseq____"^15logseq____",[87,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^Vlogseq____",98,536870919]],[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____"~u652978d9-d038-4825-a4ac-1a0c2c2a327flogseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Qlogseq____",logseq____"# [[Wurzelgleichung]]logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Flogseq____",54,536870919]],[logseq____"^15logseq____",[88,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Vlogseq____",77,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[88,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[88,logseq____"^Hlogseq____",30,536870919]],[logseq____"^15logseq____",[88,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[88,logseq____"^;logseq____",logseq____"~u652978d9-5a14-4875-bc44-81d3c31c1acalogseq____",536870919]],[logseq____"^15logseq____",[89,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____",[89,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Flogseq____",66,536870919]],[logseq____"^15logseq____",[89,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Vlogseq____",66,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[89,logseq____"^;logseq____",logseq____"~u652978d9-8c9d-442c-a6c9-e58e0673f6c1logseq____",536870919]],[logseq____"^15logseq____",[90,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____",[90,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Flogseq____",60,536870919]],[logseq____"^15logseq____",[90,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Vlogseq____",60,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",26,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",29,536870919]],[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____",26,536870919]],[logseq____"^15logseq____",[90,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[90,logseq____"^;logseq____",logseq____"~u652978d9-505a-4edf-9d98-f5348de0c2dalogseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Qlogseq____",logseq____"logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Flogseq____",80,536870919]],[logseq____"^15logseq____",[91,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Vlogseq____",103,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[91,logseq____"^;logseq____",logseq____"~u652978d9-0c0c-4270-ba7e-69aa427b8637logseq____",536870919]],[logseq____"^15logseq____",[92,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____",[92,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[92,logseq____"^Flogseq____",85,536870919]],[logseq____"^15logseq____",[92,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^Vlogseq____",85,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[92,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[92,logseq____"^;logseq____",logseq____"~u652978d9-572f-4952-adfe-8379e218b02elogseq____",536870919]],[logseq____"^15logseq____",[93,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____",[93,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Flogseq____",65,536870919]],[logseq____"^15logseq____",[93,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^Vlogseq____",60,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[93,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[93,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[93,logseq____"^;logseq____",logseq____"~u652978d9-8a3d-4644-a7a0-8f8f9c55be19logseq____",536870919]],[logseq____"^15logseq____",[94,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____",[94,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Flogseq____",72,536870919]],[logseq____"^15logseq____",[94,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^Vlogseq____",72,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[94,logseq____"^;logseq____",logseq____"~u652978d9-6ac5-49ee-be43-4814d15cbfdclogseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Qlogseq____",logseq____"# [[Binomische Formeln]]logseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Flogseq____",62,536870919]],[logseq____"^15logseq____",[95,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",34,536870919]],[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____",34,536870919]],[logseq____"^15logseq____",[95,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[95,logseq____"^;logseq____",logseq____"~u652978d9-79d6-4ea7-99f0-4fc3d6aba940logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Flogseq____",72,536870919]],[logseq____"^15logseq____",[96,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[96,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[96,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[96,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[96,logseq____"^;logseq____",logseq____"~u652978d9-4086-4aca-b48e-94ecc70b3c31logseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Qlogseq____",logseq____"$\\\\text{Alte Schreibweise}$\\n\\\\begin{align}\\n\\\\text{cosa }logseq____&\\\\text{plus }logseq____&\\\\text{cubus }logseq____&\\\\text{acq 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[[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[98,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[98,logseq____"^Flogseq____",55,536870919]],[logseq____"^15logseq____",[98,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[98,logseq____"^Vlogseq____",77,536870919]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[98,logseq____"^?logseq____",[logseq____"^ 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logseq____&= p logseq____&, 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Wissensabgleich [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[103,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[103,logseq____"^Flogseq____",104,536870919]],[logseq____"^15logseq____",[103,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[103,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[103,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[103,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[103,logseq____"^?logseq____",[logseq____"^ 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logseq____&|logseq____&-3\\\\\\\\\\n-xlogseq____&logseq____<2 logseq____&|logseq____&*-1\\\\\\\\\\nxlogseq____&logseq____>-2\\\\\\\\\\nL_1logseq____&=(-2,3)\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[134,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[134,logseq____"^Flogseq____",130,536870920]],[logseq____"^15logseq____",[134,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[134,logseq____"^Vlogseq____",130,536870920]],[logseq____"^15logseq____",[134,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[134,logseq____"^Ulogseq____",107,536870920]],[logseq____"^15logseq____",[134,logseq____"^Ulogseq____",111,536870920]],[logseq____"^15logseq____",[134,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[134,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[134,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[134,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[134,logseq____"^;logseq____",logseq____"~u652978da-f27f-4714-a0ff-70b8da1e81f8logseq____",536870920]],[logseq____"^15logseq____",[135,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 2}\\\\quad 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1a}\\\\quad x\\\\ge -2$\\n\\\\begin{align}\\n|x+2|logseq____&=x+2\\\\\\\\\\nx-3logseq____&\\\\le x+2 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[[Betrag]]logseq____",536870920]],[logseq____"^15logseq____",[138,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[138,logseq____"^Flogseq____",120,536870920]],[logseq____"^15logseq____",[138,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[138,logseq____"^Vlogseq____",116,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",111,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[138,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870920]],[logseq____"^15logseq____",[138,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[138,logseq____"^Hlogseq____",111,536870920]],[logseq____"^15logseq____",[138,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[138,logseq____"^;logseq____",logseq____"~u652978da-5963-49b0-82e0-8165f729919elogseq____",536870920]],[logseq____"^15logseq____",[139,logseq____"^Qlogseq____",logseq____"Menge 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____",[139,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[139,logseq____"^Flogseq____",133,536870920]],[logseq____"^15logseq____",[139,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[139,logseq____"^Vlogseq____",120,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[139,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[139,logseq____"^;logseq____",logseq____"~u652978da-e2ec-4c09-9a28-c7d3b11fb0aalogseq____",536870920]],[logseq____"^15logseq____",[140,logseq____"^Qlogseq____",logseq____"$\\\\{a,b,c\\\\}$logseq____",536870920]],[logseq____"^15logseq____",[140,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[140,logseq____"^Flogseq____",116,536870920]],[logseq____"^15logseq____",[140,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[140,logseq____"^Vlogseq____",116,536870920]],[logseq____"^15logseq____",[140,logseq____"^Ulogseq____",112,536870920]],[logseq____"^15logseq____",[140,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[140,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[140,logseq____"^;logseq____",logseq____"~u652978da-766f-43ef-a922-ab3959daf508logseq____",536870920]],[logseq____"^15logseq____",[141,logseq____"^Qlogseq____",logseq____"### 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logseq____",logseq____"^18logseq____",3],536870920]],[logseq____"^15logseq____",[141,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[141,logseq____"^Hlogseq____",107,536870920]],[logseq____"^15logseq____",[141,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[141,logseq____"^;logseq____",logseq____"~u652978da-3024-42f5-a245-4fa978a06332logseq____",536870920]],[logseq____"^15logseq____",[142,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 2}\\\\quad x\\\\ge3$\\n\\\\begin{align}\\n|x-3|logseq____&=-x+3\\\\\\\\\\nx-3logseq____&logseq____<5 logseq____&|logseq____&+3\\\\\\\\\\nxlogseq____&logseq____<8\\\\\\\\\\nL_2logseq____&=[3,8)\\\\\\\\\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[142,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[142,logseq____"^Flogseq____",134,536870920]],[logseq____"^15logseq____",[142,logseq____"^Xlogseq____",112,536870920]],[logseq____"^15logseq____",[142,logseq____"^Vlogseq____",130,536870920]],[logseq____"^15logseq____",[142,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[142,logseq____"^Ulogseq____",107,536870920]],[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____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[142,logseq____"^;logseq____",logseq____"~u652978da-f59b-4724-afb2-ba5e832b1164logseq____",536870920]],[logseq____"^15logseq____",[143,logseq____"^Qlogseq____",logseq____"Kombinieren:\\n\\\\begin{align}\\nL=L_1\\\\cup 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logseq____",logseq____"^18logseq____",3],536870920]],[logseq____"^15logseq____",[144,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[144,logseq____"^Hlogseq____",107,536870920]],[logseq____"^15logseq____",[144,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[144,logseq____"^;logseq____",logseq____"~u652978da-ea66-4ed4-8b68-ee8932268320logseq____",536870920]],[logseq____"^15logseq____",[145,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 3}\\\\quad xlogseq____>3$\\n\\\\begin{align}\\nx+5logseq____&logseq____>0\\\\\\\\\\n|x+5|logseq____&=x+5\\\\\\\\\\nx-3logseq____&\\\\ge0\\\\\\\\\\n|x-3|logseq____&=x-3\\\\\\\\\\nx+5logseq____&logseq____<2(x-3)\\\\\\\\\\nx+5logseq____&logseq____<2x-3 logseq____&|logseq____& -2x-5\\\\\\\\\\n-xlogseq____&logseq____<-11 logseq____&|logseq____& 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q$|\\n|-|-|-|-|-|-|\\n|f|f||w|w|w|w|\\n|f|w||w|f|f|f|\\n|w|f||f|w|f|f|\\n|w|w||w|w|w|w|logseq____",536870922]],[logseq____"^15logseq____",[187,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[187,logseq____"^Flogseq____",206,536870922]],[logseq____"^15logseq____",[187,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[187,logseq____"^Vlogseq____",197,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",162,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[187,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[187,logseq____"^Hlogseq____",162,536870922]],[logseq____"^15logseq____",[187,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[187,logseq____"^;logseq____",logseq____"~u652978da-e27c-4c4e-9b5a-b80e17f2ff8alogseq____",536870922]],[logseq____"^15logseq____",[188,logseq____"^Qlogseq____",logseq____"## [[Verknüpfung von Aussagen]]logseq____",536870922]],[logseq____"^15logseq____",[188,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[188,logseq____"^Flogseq____",191,536870922]],[logseq____"^15logseq____",[188,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[188,logseq____"^Vlogseq____",198,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[188,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[188,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870922]],[logseq____"^15logseq____",[188,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[188,logseq____"^Hlogseq____",178,536870922]],[logseq____"^15logseq____",[188,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[188,logseq____"^;logseq____",logseq____"~u652978da-b8fa-4f1b-bfbf-4355a7b1a6eelogseq____",536870922]],[logseq____"^15logseq____",[189,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____",[189,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[189,logseq____"^Flogseq____",188,536870922]],[logseq____"^15logseq____",[189,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[189,logseq____"^Vlogseq____",188,536870922]],[logseq____"^15logseq____",[189,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[189,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[189,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[189,logseq____"^Hlogseq____",174,536870922]],[logseq____"^15logseq____",[189,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[189,logseq____"^;logseq____",logseq____"~u652978da-10f5-4eb0-bc43-61573e519566logseq____",536870922]],[logseq____"^15logseq____",[190,logseq____"^Qlogseq____",logseq____"### [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[190,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[190,logseq____"^Flogseq____",197,536870922]],[logseq____"^15logseq____",[190,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[190,logseq____"^Vlogseq____",188,536870922]],[logseq____"^15logseq____",[190,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[190,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[190,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[190,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[190,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[190,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[190,logseq____"^Hlogseq____",168,536870922]],[logseq____"^15logseq____",[190,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[190,logseq____"^;logseq____",logseq____"~u652978da-8c22-4205-a0a2-00d77513a432logseq____",536870922]],[logseq____"^15logseq____",[191,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____",[191,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[191,logseq____"^Flogseq____",186,536870922]],[logseq____"^15logseq____",[191,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[191,logseq____"^Vlogseq____",198,536870922]],[logseq____"^15logseq____",[191,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[191,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[191,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[191,logseq____"^;logseq____",logseq____"~u652978da-ca44-4cae-8206-b7af2d645011logseq____",536870922]],[logseq____"^15logseq____",[192,logseq____"^Qlogseq____",logseq____"Formel, die konstant $w$ istlogseq____",536870922]],[logseq____"^15logseq____",[192,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[192,logseq____"^Flogseq____",190,536870922]],[logseq____"^15logseq____",[192,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[192,logseq____"^Vlogseq____",190,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[192,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[192,logseq____"^;logseq____",logseq____"~u652978da-2761-467e-9bba-d96fac88510clogseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Qlogseq____",logseq____"# Büchertiplogseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Flogseq____",159,536870922]],[logseq____"^15logseq____",[193,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[193,logseq____"^Vlogseq____",159,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[193,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[193,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[193,logseq____"^17logseq____",false,536870922]],[logseq____"^15logseq____",[193,logseq____"^;logseq____",logseq____"~u652978da-ad57-49bd-992d-e2c266efd76alogseq____",536870922]],[logseq____"^15logseq____",[194,logseq____"^Qlogseq____",logseq____"[Variable]([[Variablen]]), die den Wert $w$ oder $f$ annimmtlogseq____",536870922]],[logseq____"^15logseq____",[194,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[194,logseq____"^Flogseq____",201,536870922]],[logseq____"^15logseq____",[194,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[194,logseq____"^Vlogseq____",201,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",163,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",170,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[194,logseq____"^Hlogseq____",170,536870922]],[logseq____"^15logseq____",[194,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[194,logseq____"^;logseq____",logseq____"~u652978da-7c7b-48fc-837d-847dba68bd89logseq____",536870922]],[logseq____"^15logseq____",[195,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____",[195,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[195,logseq____"^Flogseq____",184,536870922]],[logseq____"^15logseq____",[195,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[195,logseq____"^Vlogseq____",188,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",166,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",171,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",173,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",177,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[195,logseq____"^Hlogseq____",166,536870922]],[logseq____"^15logseq____",[195,logseq____"^Hlogseq____",171,536870922]],[logseq____"^15logseq____",[195,logseq____"^Hlogseq____",173,536870922]],[logseq____"^15logseq____",[195,logseq____"^Hlogseq____",177,536870922]],[logseq____"^15logseq____",[195,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[195,logseq____"^;logseq____",logseq____"~u652978da-c6c0-4de5-ba76-067f200176e9logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Qlogseq____",logseq____"C. Meinel, M. Mundhenk, [[\\logseq____"Mathematische Grundlagen der Informatik\\logseq____", 5. Auflage 2011]]logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Flogseq____",193,536870922]],[logseq____"^15logseq____",[196,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[196,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",167,536870922]],[logseq____"^15logseq____",[196,logseq____"^Hlogseq____",167,536870922]],[logseq____"^15logseq____",[196,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[196,logseq____"^;logseq____",logseq____"~u652978da-24ee-4142-a8cb-05067d3ac608logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Qlogseq____",logseq____"### [[Wahrheitswerteverlauf]] einer [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Flogseq____",183,536870922]],[logseq____"^15logseq____",[197,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[197,logseq____"^Vlogseq____",188,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",162,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[197,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[197,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[197,logseq____"^Hlogseq____",162,536870922]],[logseq____"^15logseq____",[197,logseq____"^Hlogseq____",176,536870922]],[logseq____"^15logseq____",[197,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[197,logseq____"^;logseq____",logseq____"~u652978da-4438-4716-b7f9-0760fec70081logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Qlogseq____",logseq____"# 1. [[Aussagen]]logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Flogseq____",193,536870922]],[logseq____"^15logseq____",[198,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[198,logseq____"^Vlogseq____",159,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[198,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[198,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[198,logseq____"^Hlogseq____",174,536870922]],[logseq____"^15logseq____",[198,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[198,logseq____"^;logseq____",logseq____"~u652978da-8b54-470e-a86d-c01533144650logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(p\\\\rightarrow q)logseq____&\\\\equiv((\\\\neg q)\\\\rightarrow(\\\\neg p))\\\\\\\\\\n(p\\\\land(p\\\\rightarrow q)) logseq____&\\\\rightarrow q logseq____&\\\\quad \\\\text{Tantologie}\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Flogseq____",202,536870922]],[logseq____"^15logseq____",[199,logseq____"^Xlogseq____",159,536870922]],[logseq____"^15logseq____",[199,logseq____"^Vlogseq____",202,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[199,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[199,logseq____"^;logseq____",logseq____"~u652978da-9bd7-408a-89ff-c72fb59c264alogseq____",536870922]],[logseq____"^15logseq____",[200,logseq____"^Qlogseq____",logseq____"Formeln $p,q$ [[logisch äquivalent]], wenn sie den gleichen [[Wahrheitswerteverlauf]] haben.\\n[[Schreibweise]]: $p\\\\equiv 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[[assoziativ]] [[kommutativ]] [[idenpotent]] [[wechselseitig distributiv]] [[distributiv]] [[de Morgansche 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$x$ aus $\\\\Reals$ beliebig. Setze $y:=x+1$ (ist aus $\\\\Reals$). Dann ist $xlogseq____<x+1=y$ also $xlogseq____<y$. 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= ↯, 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([Beispiel]([[Beispiele]])):logseq____",536870923]],[logseq____"^15logseq____",[241,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[241,logseq____"^Flogseq____",272,536870923]],[logseq____"^15logseq____",[241,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[241,logseq____"^Vlogseq____",259,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[241,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[241,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[241,logseq____"^;logseq____",logseq____"~u652978da-d8e9-4238-b06d-f23d482a497dlogseq____",536870923]],[logseq____"^15logseq____",[242,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 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$\\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 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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____",258,536870923]],[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____"~u652978da-9627-477c-84e2-738c0a84c5c6logseq____",536870923]],[logseq____"^15logseq____",[248,logseq____"^Qlogseq____",logseq____"[Gleichartige]([[Gleichartig]]) [[Quantoren]] dürfen vertauscht 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mit [[Variablen]] $x_1,\\\\cdots,x_n$, der bei Ersetzung jedes $x_j$ durch ein [Objekt]([[Objekt (Logik)]]) aus $U_j$ stets zu einer [Aussage]([[Aussagen]]) 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= Beweisende, auch [q.e.d.]([[Q.E.D.]]), [[w.z.b.w.]], QED, 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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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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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x:\\\\exists y:p(x,y)\\\\equiv\\\\exists y:\\\\exists x:p(x,y)\\\\\\\\\\nlogseq____&[\\\\forall x_1:\\\\forall x_2: \\\\cdots:\\\\forall x_n:p(x_1,x_2,\\\\cdots,x_n)\\\\notag\\\\\\\\\\nlogseq____&\\\\equiv\\\\forall_{x_{\\\\pi(1)}}:\\\\forall_{x_{\\\\pi(1)}}:\\\\cdots:\\\\forall_{x_{\\\\pi(1)}}:p(x_1,x_2,\\\\cdots,x_n)\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[256,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[256,logseq____"^Flogseq____",248,536870923]],[logseq____"^15logseq____",[256,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[256,logseq____"^Vlogseq____",248,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[256,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[256,logseq____"^;logseq____",logseq____"~u652978da-4fbf-4a48-9ba2-d2fce807a8e5logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Qlogseq____",logseq____"[[Aussageform]] (AF) über [Universen]([[Universen (Logik)]]) $U_1,\\\\cdots,U_n$:logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Flogseq____",244,536870923]],[logseq____"^15logseq____",[257,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[257,logseq____"^Vlogseq____",244,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",232,536870923]],[logseq____"^15logseq____",[257,logseq____"^Hlogseq____",228,536870923]],[logseq____"^15logseq____",[257,logseq____"^Hlogseq____",232,536870923]],[logseq____"^15logseq____",[257,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[257,logseq____"^;logseq____",logseq____"~u652978da-19a5-4587-87e5-b7aead08363flogseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Qlogseq____",logseq____"# [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Flogseq____",217,536870923]],[logseq____"^15logseq____",[258,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[258,logseq____"^Vlogseq____",217,536870923]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[258,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[258,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[258,logseq____"^Hlogseq____",210,536870923]],[logseq____"^15logseq____",[258,logseq____"^17logseq____",false,536870923]],[logseq____"^15logseq____",[258,logseq____"^;logseq____",logseq____"~u652978da-29da-4169-aea0-1f2b840d48c4logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Qlogseq____",logseq____"## [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Flogseq____",238,536870923]],[logseq____"^15logseq____",[259,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[259,logseq____"^Vlogseq____",244,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[259,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870923]],[logseq____"^15logseq____",[259,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[259,logseq____"^Hlogseq____",210,536870923]],[logseq____"^15logseq____",[259,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[259,logseq____"^;logseq____",logseq____"~u652978da-bf17-461e-aff4-286d56eaf7f6logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Qlogseq____",logseq____"wobei $\\\\pi$ eine \\logseq____"[[Permutation]]\\logseq____" der Zahl $(1,2,\\\\cdots,n)$ istlogseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Flogseq____",256,536870923]],[logseq____"^15logseq____",[260,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[260,logseq____"^Vlogseq____",256,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[260,logseq____"^Hlogseq____",224,536870923]],[logseq____"^15logseq____",[260,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[260,logseq____"^;logseq____",logseq____"~u652978da-3a75-4990-8cd6-b8a82bc24973logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Qlogseq____",logseq____"see: [[Große Azzoziativgesetze]] [[Große de Morgansche Regeln]] [[Negationsregeln]]logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Flogseq____",272,536870923]],[logseq____"^15logseq____",[261,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[261,logseq____"^Vlogseq____",272,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",210,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",213,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",225,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",230,536870923]],[logseq____"^15logseq____",[261,logseq____"^Hlogseq____",213,536870923]],[logseq____"^15logseq____",[261,logseq____"^Hlogseq____",225,536870923]],[logseq____"^15logseq____",[261,logseq____"^Hlogseq____",230,536870923]],[logseq____"^15logseq____",[261,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[261,logseq____"^;logseq____",logseq____"~u652978da-8bbe-4f4e-a679-86a1c8ccf6b3logseq____",536870923]],[logseq____"^15logseq____",[262,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____",[262,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Flogseq____",245,536870923]],[logseq____"^15logseq____",[262,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[262,logseq____"^Vlogseq____",245,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",221,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[262,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[262,logseq____"^;logseq____",logseq____"~u652978da-fea0-42a4-9142-f6aa1a30087elogseq____",536870923]],[logseq____"^15logseq____",[263,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____",[263,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Flogseq____",251,536870923]],[logseq____"^15logseq____",[263,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[263,logseq____"^Vlogseq____",251,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",221,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[263,logseq____"^Hlogseq____",221,536870923]],[logseq____"^15logseq____",[263,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[263,logseq____"^;logseq____",logseq____"~u652978da-2de0-406b-ab99-c852859cee7clogseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Flogseq____",256,536870923]],[logseq____"^15logseq____",[264,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[264,logseq____"^Vlogseq____",248,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",35,536870923]],[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____",35,536870923]],[logseq____"^15logseq____",[264,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[264,logseq____"^;logseq____",logseq____"~u652978da-fa92-40b1-9f43-fad3770c90a2logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Qlogseq____",logseq____"Für 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____",[265,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Flogseq____",273,536870923]],[logseq____"^15logseq____",[265,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[265,logseq____"^Vlogseq____",244,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[265,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[265,logseq____"^;logseq____",logseq____"~u652978da-c56f-426c-9a9d-9e259fe68241logseq____",536870923]],[logseq____"^15logseq____",[266,logseq____"^Qlogseq____",logseq____"Endliches Universum $U$ mit Objekten $O_1,\\\\cdots,O_m$logseq____",536870923]],[logseq____"^15logseq____",[266,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[266,logseq____"^Flogseq____",265,536870923]],[logseq____"^15logseq____",[266,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[266,logseq____"^Vlogseq____",244,536870923]],[logseq____"^15logseq____",[266,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[266,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[266,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[266,logseq____"^;logseq____",logseq____"~u652978da-399a-44c4-80b4-a85a01e60e3alogseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Qlogseq____",logseq____"#FIXMElogseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Flogseq____",238,536870923]],[logseq____"^15logseq____",[267,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[267,logseq____"^Vlogseq____",238,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",42,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[267,logseq____"^Hlogseq____",42,536870923]],[logseq____"^15logseq____",[267,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[267,logseq____"^;logseq____",logseq____"~u652978da-675f-4bbf-a78f-c046b9aca568logseq____",536870923]],[logseq____"^15logseq____",[268,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____",[268,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[268,logseq____"^Flogseq____",254,536870923]],[logseq____"^15logseq____",[268,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[268,logseq____"^Vlogseq____",254,536870923]],[logseq____"^15logseq____",[268,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[268,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[268,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[268,logseq____"^;logseq____",logseq____"~u652978da-b3d6-4d38-a95a-b9c0025fc729logseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Qlogseq____",logseq____"Verschiedene [[Quantoren]] dürfen **nicht** vertauscht werdenlogseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Flogseq____",260,536870923]],[logseq____"^15logseq____",[269,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[269,logseq____"^Vlogseq____",256,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[269,logseq____"^Hlogseq____",222,536870923]],[logseq____"^15logseq____",[269,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[269,logseq____"^;logseq____",logseq____"~u652978da-c7e0-46a8-afa3-e1ffc82ca69elogseq____",536870923]],[logseq____"^15logseq____",[270,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____",[270,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[270,logseq____"^Flogseq____",266,536870923]],[logseq____"^15logseq____",[270,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[270,logseq____"^Vlogseq____",244,536870923]],[logseq____"^15logseq____",[270,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[270,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[270,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[270,logseq____"^;logseq____",logseq____"~u652978da-5ef2-4190-9b18-4930122279f2logseq____",536870923]],[logseq____"^15logseq____",[271,logseq____"^Qlogseq____",logseq____"see: [[Allquantor]] [[Existenzquantor]]logseq____",536870923]],[logseq____"^15logseq____",[271,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[271,logseq____"^Flogseq____",270,536870923]],[logseq____"^15logseq____",[271,logseq____"^Xlogseq____",217,536870923]],[logseq____"^15logseq____",[271,logseq____"^Vlogseq____",270,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",229,536870923]],[logseq____"^15logseq____",[271,logseq____"^Hlogseq____",223,536870923]],[logseq____"^15logseq____",[271,logseq____"^Hlogseq____",229,536870923]],[logseq____"^15logseq____",[271,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[271,logseq____"^;logseq____",logseq____"~u652978da-0875-4684-b510-e07639365e7clogseq____",536870923]],[logseq____"^15logseq____",[272,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall 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