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Falschlogseq____",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____",78,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[54,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[54,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[54,logseq____"^;logseq____",logseq____"~u652fe1d8-0834-411a-8bf4-a861a5990ff2logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\sqrt{x^4+6x^2y+9y^2}logseq____&=\\\\sqrt{(x^2+3y)^2}=x^2+3y\\\\\\\\\\n\\\\sqrt {x + \\\\sqrt{x^2-y^2}}\\\\sqrt{x-\\\\sqrt{x^2-y^2}}logseq____&=\\\\sqrt{\\\\left(x + \\\\sqrt{x^2-y^2}\\\\right)\\\\left(x-\\\\sqrt{x^2-y^2}\\\\right)}\\\\notag\\\\\\\\\\nlogseq____&=\\\\sqrt{x^2-(x^2-y^2)}\\\\notag\\\\\\\\\\nlogseq____&=\\\\sqrt{y^2}=\\\\left|y\\\\right|\\\\\\\\\\n\\\\sqrt[5]{a\\\\sqrt[3]a}=\\\\sqrt[5]{\\\\sqrt[3]{a^3}\\\\sqrt[3]a}\\nlogseq____&=\\\\sqrt[5]{\\\\sqrt[3]{a^4}}\\n=\\\\sqrt[5]{a^\\\\frac43}\\n=a^{\\\\frac43\\\\frac15}=a^\\\\frac4{15}\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Flogseq____",50,536870919]],[logseq____"^15logseq____",[55,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^Vlogseq____",50,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[55,logseq____"^;logseq____",logseq____"~u652fe1d8-4dd3-41b7-b158-778db5911fd2logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Qlogseq____",logseq____"## [[pq-Formel]]logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Flogseq____",99,536870919]],[logseq____"^15logseq____",[56,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Vlogseq____",104,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[56,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[56,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[56,logseq____"^Hlogseq____",43,536870919]],[logseq____"^15logseq____",[56,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[56,logseq____"^;logseq____",logseq____"~u652fe1d7-3ea5-494f-b3e5-45dd398bf487logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Qlogseq____",logseq____"## [[Binomialkoeffizient]]logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Flogseq____",78,536870919]],[logseq____"^15logseq____",[57,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[57,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[57,logseq____"^Hlogseq____",32,536870919]],[logseq____"^15logseq____",[57,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[57,logseq____"^;logseq____",logseq____"~u652fe1d8-f08e-42ec-84ec-7e0ccc4f09eelogseq____",536870919]],[logseq____"^15logseq____",[58,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____",[58,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Flogseq____",48,536870919]],[logseq____"^15logseq____",[58,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^Vlogseq____",56,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[58,logseq____"^Hlogseq____",41,536870919]],[logseq____"^15logseq____",[58,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[58,logseq____"^;logseq____",logseq____"~u652fe1d7-5f6e-437b-8bc7-2ef1cff4981blogseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Qlogseq____",logseq____"# [[Potenzen]] und [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Flogseq____",72,536870919]],[logseq____"^15logseq____",[59,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[59,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[59,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[59,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[59,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[59,logseq____"^;logseq____",logseq____"~u652fe1d8-25ec-4a85-b7af-744d5cd192bflogseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Flogseq____",96,536870919]],[logseq____"^15logseq____",[60,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[60,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[60,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[60,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[60,logseq____"^;logseq____",logseq____"~u652fe1d8-e89c-41c1-a728-c649a3405199logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Qlogseq____",logseq____"# [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Flogseq____",59,536870919]],[logseq____"^15logseq____",[61,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^Vlogseq____",59,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[61,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[61,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[61,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[61,logseq____"^;logseq____",logseq____"~u652fe1d8-91dc-4973-a77e-cd75b12e5244logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Qlogseq____",logseq____"https://www.wolframalpha.comlogseq____",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____",66,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[62,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[62,logseq____"^;logseq____",logseq____"~u652fe1d8-9411-4aa9-8b0f-8632bb2dd466logseq____",536870919]],[logseq____"^15logseq____",[63,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____",[63,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Flogseq____",58,536870919]],[logseq____"^15logseq____",[63,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^Vlogseq____",56,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[63,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[63,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[63,logseq____"^;logseq____",logseq____"~u652fe1d7-9d19-41e1-80fd-6ba23228c4b2logseq____",536870919]],[logseq____"^15logseq____",[64,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____",[64,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Flogseq____",102,536870919]],[logseq____"^15logseq____",[64,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^Vlogseq____",102,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[64,logseq____"^;logseq____",logseq____"~u652fe1d8-2f5d-4cd0-92bc-fcf4d0504c18logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Qlogseq____",logseq____"# [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Flogseq____",61,536870919]],[logseq____"^15logseq____",[65,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Vlogseq____",59,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[65,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[65,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[65,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[65,logseq____"^;logseq____",logseq____"~u652fe1d8-3223-4c3c-9c09-063fb81890f8logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Qlogseq____",logseq____"[[Wolframalpha]]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____",96,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[66,logseq____"^Hlogseq____",46,536870919]],[logseq____"^15logseq____",[66,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[66,logseq____"^;logseq____",logseq____"~u652fe1d8-9892-44a7-9de1-ba261a59b929logseq____",536870919]],[logseq____"^15logseq____",[67,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____",[67,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Flogseq____",47,536870919]],[logseq____"^15logseq____",[67,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Vlogseq____",47,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[67,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[67,logseq____"^;logseq____",logseq____"~u652fe1d7-218b-4a5a-a813-ded68cacc9b9logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Flogseq____",102,536870919]],[logseq____"^15logseq____",[68,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[68,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[68,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[68,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[68,logseq____"^;logseq____",logseq____"~u652fe1d8-a1ba-465a-b8fa-e89642aba357logseq____",536870919]],[logseq____"^15logseq____",[69,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____",[69,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Flogseq____",77,536870919]],[logseq____"^15logseq____",[69,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^Vlogseq____",68,536870919]],[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____"~u652fe1d8-ac6b-4c8a-91c0-5789047dac2clogseq____",536870919]],[logseq____"^15logseq____",[70,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____",[70,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Flogseq____",50,536870919]],[logseq____"^15logseq____",[70,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^Vlogseq____",65,536870919]],[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____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[70,logseq____"^;logseq____",logseq____"~u652fe1d8-af2c-45f8-aa27-4e70c52922d6logseq____",536870919]],[logseq____"^15logseq____",[71,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____",[71,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Flogseq____",103,536870919]],[logseq____"^15logseq____",[71,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Vlogseq____",103,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[71,logseq____"^;logseq____",logseq____"~u652fe1d8-13d1-4a3c-b0eb-426396245365logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Qlogseq____",logseq____"# [[Binomische Formeln]]logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Flogseq____",89,536870919]],[logseq____"^15logseq____",[72,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",34,536870919]],[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____",34,536870919]],[logseq____"^15logseq____",[72,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[72,logseq____"^;logseq____",logseq____"~u652fe1d8-28a7-45bd-a80f-c412fd3cc6calogseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Qlogseq____",logseq____"#FIXME $\\\\sin^{-2}x+cos^2y=1,r$ ist falschlogseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Flogseq____",81,536870919]],[logseq____"^15logseq____",[73,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^Vlogseq____",81,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[73,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[73,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[73,logseq____"^;logseq____",logseq____"~u652fe1d8-349c-4740-b1b3-eef2d7f3db9elogseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche quadratische Gleichung hat die Lösung}$\\n$x_1=3\\\\quad x_2=4$\\n\\\\begin{align}\\nx_1+x_2 = 7 logseq____&= p logseq____&, x_1x_2=12=q\\\\\\\\\\nx^2+px+qlogseq____&=0\\\\\\\\\\nx^2-7x+12logseq____&=0\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Flogseq____",94,536870919]],[logseq____"^15logseq____",[74,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^Vlogseq____",94,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",29,536870919]],[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____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[74,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[74,logseq____"^;logseq____",logseq____"~u652fe1d7-e067-4e5a-b1ef-f7cfca8a3e83logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Qlogseq____",logseq____"$\\\\text{Alte Schreibweise}$\\n\\\\begin{align}\\n\\\\text{cosa }logseq____&\\\\text{plus }logseq____&\\\\text{cubus }logseq____&\\\\text{acq }logseq____&6\\\\\\\\\\nxlogseq____&+logseq____&x^3logseq____&=logseq____&6\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Flogseq____",79,536870919]],[logseq____"^15logseq____",[75,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^Vlogseq____",61,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[75,logseq____"^;logseq____",logseq____"~u652fe1d8-eeae-4fc1-8bae-d1da0caeeb86logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Qlogseq____",logseq____"# [[Wurzelgleichung]]logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Flogseq____",104,536870919]],[logseq____"^15logseq____",[76,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Vlogseq____",59,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[76,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[76,logseq____"^Hlogseq____",30,536870919]],[logseq____"^15logseq____",[76,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[76,logseq____"^;logseq____",logseq____"~u652fe1d7-e3bb-4cfc-9cc6-b1776ba94a3elogseq____",536870919]],[logseq____"^15logseq____",[77,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____",[77,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Flogseq____",68,536870919]],[logseq____"^15logseq____",[77,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^Vlogseq____",68,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[77,logseq____"^;logseq____",logseq____"~u652fe1d8-fcc4-428b-9e7a-72db00d65bfflogseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Flogseq____",90,536870919]],[logseq____"^15logseq____",[78,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[78,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[78,logseq____"^;logseq____",logseq____"~u652fe1d8-4c60-4f70-935a-0f3fa4faa2d3logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Flogseq____",84,536870919]],[logseq____"^15logseq____",[79,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Vlogseq____",61,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[79,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[79,logseq____"^;logseq____",logseq____"~u652fe1d8-7300-4505-99f4-300a51856c4flogseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Qlogseq____",logseq____"Einsetzen, um zu überprüfen, ob man eine Fallunterscheidung brauchtlogseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Flogseq____",52,536870919]],[logseq____"^15logseq____",[80,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[80,logseq____"^;logseq____",logseq____"~u652fe1d7-05e6-4275-bb58-c717ca74da94logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche Gleichungen sind richtig?}$\\n\\\\begin{align}\\n\\\\tan x logseq____&= \\\\frac{\\\\sin x}{\\\\cos x} logseq____&, \\\\cos x \\\\ne 0\\\\\\\\\\n\\\\sin^{-2}x-\\\\cos^2xlogseq____&=1logseq____&,f\\\\\\\\\\n\\\\sin x = \\\\cos x logseq____&= 1 logseq____&, f\\\\\\\\\\n\\\\sin^{-2}x+cos^2ylogseq____&=1logseq____&,r\\\\\\\\\\n1+tan^2xlogseq____&=\\\\frac1{\\\\cos^2x}logseq____&,\\\\cos x\\\\ne0\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Flogseq____",49,536870919]],[logseq____"^15logseq____",[81,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^Vlogseq____",89,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[81,logseq____"^;logseq____",logseq____"~u652fe1d8-45c5-423f-8bf5-cd7cc59f46a4logseq____",536870919]],[logseq____"^15logseq____",[82,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____",[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____",72,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[82,logseq____"^;logseq____",logseq____"~u652fe1d8-4ae0-44da-abe0-456f9ef2d3c3logseq____",536870919]],[logseq____"^15logseq____",[83,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____",[83,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Flogseq____",91,536870919]],[logseq____"^15logseq____",[83,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^Vlogseq____",91,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[83,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[83,logseq____"^;logseq____",logseq____"~u652fe1d7-af72-4044-833e-20615ddf257elogseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\na^nlogseq____&=\\\\underbrace{a*a*\\\\cdots*n}_n logseq____& a,blogseq____>0\\\\\\\\\\na^2b^2logseq____&=(ab)^n\\\\\\\\\\n\\\\frac{a^n}{b^n}logseq____&=\\\\left(\\\\frac ab\\\\right)^n\\\\\\\\\\na^na^mlogseq____&=a^{n+m}\\\\\\\\\\n\\\\frac1{a^n}logseq____&=a^{-n}\\\\\\\\\\n(a^n)^mlogseq____&=a^{nm}\\\\\\\\\\na^0logseq____&=1\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Flogseq____",61,536870919]],[logseq____"^15logseq____",[84,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^Vlogseq____",61,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[84,logseq____"^;logseq____",logseq____"~u652fe1d8-e5ab-4085-a527-0bbe9e728caflogseq____",536870919]],[logseq____"^15logseq____",[85,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____",[85,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Flogseq____",65,536870919]],[logseq____"^15logseq____",[85,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^Vlogseq____",65,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[85,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[85,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[85,logseq____"^;logseq____",logseq____"~u652fe1d8-1fac-4515-b085-4c2e77a074a6logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Qlogseq____",logseq____"## [[Pascalsches Dreieck]]logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Flogseq____",82,536870919]],[logseq____"^15logseq____",[86,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Vlogseq____",72,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[86,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[86,logseq____"^Hlogseq____",28,536870919]],[logseq____"^15logseq____",[86,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[86,logseq____"^;logseq____",logseq____"~u652fe1d8-e31e-4964-be24-1627ac9dc580logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Qlogseq____",logseq____"logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Flogseq____",93,536870919]],[logseq____"^15logseq____",[87,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^Vlogseq____",60,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[87,logseq____"^;logseq____",logseq____"~u652fe1d8-bd54-45ae-8743-883a14c49348logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn ergibtlogseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Flogseq____",55,536870919]],[logseq____"^15logseq____",[88,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Vlogseq____",50,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[88,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[88,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[88,logseq____"^;logseq____",logseq____"~u652fe1d8-5b9d-4167-8c39-5832603e7a7elogseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Trigonometrie]]logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Flogseq____",103,536870919]],[logseq____"^15logseq____",[89,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[89,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[89,logseq____"^Hlogseq____",36,536870919]],[logseq____"^15logseq____",[89,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[89,logseq____"^;logseq____",logseq____"~u652fe1d8-96fd-4ac5-8fa7-dde8cd3480e9logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Qlogseq____",logseq____"\\\\begin{array}{c} (a+b)^0logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&1 \\\\\\\\\\n(a+b)^1logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&1 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^2logseq____&logseq____&logseq____&logseq____&logseq____&logseq____& 1 logseq____&logseq____& 2 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^3logseq____&logseq____&logseq____&logseq____&logseq____&1 logseq____&logseq____& 3 logseq____&logseq____& 3 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^4logseq____&logseq____&logseq____&logseq____& 1 logseq____&logseq____& 4 logseq____&logseq____& 6 logseq____&logseq____& 4 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^5logseq____&logseq____&logseq____& 1 logseq____&logseq____& 5 logseq____&logseq____& 10 logseq____&logseq____& 10 logseq____&logseq____& 5 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^6logseq____&logseq____& 1 logseq____&logseq____& 6 logseq____&logseq____& 15 logseq____&logseq____& 20 logseq____&logseq____& 15 logseq____&logseq____& 6 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^7logseq____&1 logseq____&logseq____& 7 logseq____&logseq____&21 logseq____&logseq____& 35 logseq____&logseq____& 35 logseq____&logseq____& 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}$[[Gleichungen]]$\\\\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____",[93,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Flogseq____",60,536870919]],[logseq____"^15logseq____",[93,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^Vlogseq____",60,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",33,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^Hlogseq____",33,536870919]],[logseq____"^15logseq____",[93,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[93,logseq____"^;logseq____",logseq____"~u652fe1d8-b4d6-4711-9986-80e2cc2eb84alogseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Flogseq____",58,536870919]],[logseq____"^15logseq____",[94,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^Vlogseq____",58,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[94,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[94,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[94,logseq____"^;logseq____",logseq____"~u652fe1d7-7003-4aeb-97b3-eca56186e0b5logseq____",536870919]],[logseq____"^15logseq____",[95,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 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[[Links]]logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Flogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[96,logseq____"^?logseq____",[logseq____"^ 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x)^5*(1-\\\\sqrt 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nn=a^1=a\\\\\\\\\\n\\\\sqrt[n]{\\\\frac ab} logseq____&= \\\\frac{\\\\sqrt[n]a}{\\\\sqrt[n]b}\\\\\\\\\\n\\\\left(\\\\sqrt[n]a\\\\right)^mlogseq____&=a^\\\\frac 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1} \\\\quad xlogseq____<-5$\\n\\\\begin{align}\\nx+5logseq____&logseq____<0\\\\\\\\\\n|x+5|logseq____&=-(x+5)=-x-5\\\\\\\\\\nx-3logseq____&logseq____<0\\\\\\\\\\n|x-3|logseq____&=-(x-3)=-x+3\\\\\\\\\\n-x-5logseq____&logseq____< 2(-x+3)\\\\\\\\\\n-x-5logseq____&logseq____<-2x+6 logseq____&|logseq____&+5+2x\\\\\\\\\\nxlogseq____<11\\\\\\\\\\n\\\\end{align}\\n\\\\begin{align}\\nL_1=\\\\left\\\\{x\\\\in\\\\Reals\\\\Big|xlogseq____<-5,x\\\\le11\\\\right\\\\}=(-\\\\infty,-5)\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[119,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[119,logseq____"^Flogseq____",140,536870920]],[logseq____"^15logseq____",[119,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[119,logseq____"^Vlogseq____",140,536870920]],[logseq____"^15logseq____",[119,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[119,logseq____"^Ulogseq____",108,536870920]],[logseq____"^15logseq____",[119,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[119,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[119,logseq____"^Ulogseq____",115,536870920]],[logseq____"^15logseq____",[119,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[119,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[119,logseq____"^;logseq____",logseq____"~u652fe1d8-1973-44da-aec8-60d94a45eb92logseq____",536870920]],[logseq____"^15logseq____",[120,logseq____"^Qlogseq____",logseq____"[Menge]([[Mengen]]) von [Punkten]([[Punkte]]) $(x,y)$ in der [Ebene]([[Ebenen]]) die den gleichen [[Abstand]] r ([[Radius]]) zu einem festen [Punkt]([[Punkte]]) 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aller rellen Zahlen zwischen 4 und $\\\\infty$ ohne 4 und 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aller reellen Zahlen zwischen 1 und 4 inklusive 1 und 4\\n\\\\begin{align}\\\\left\\\\{x\\\\in\\\\Reals\\\\middle|-1\\\\le x\\\\le4\\\\right\\\\}=:\\\\underbrace{[-1,4]}_\\\\text{Abgeschlossenes 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der Gleichung }x^2+\\\\underbrace{4}_p x-\\\\underbrace{12}_q=0$\\n\\\\begin{align}\\nx_{1,2}logseq____&=\\\\frac{-p}2\\\\pm\\\\sqrt{\\\\frac{p^2}4-q}\\\\\\\\\\nlogseq____&=-2\\\\pm\\\\sqrt{\\\\frac{16}4-(-12)}\\\\\\\\\\nlogseq____&=-2\\\\pm4\\\\\\\\\\nLlogseq____&=\\\\left\\\\{\\\\underbrace{x\\\\in\\\\Reals}_\\\\text{Name}\\\\middle|\\\\underbrace{x^2+4x-12=0}_\\\\text{Eigenschaft}\\\\right\\\\}\\\\\\\\\\nLlogseq____&=\\\\left\\\\{2,-6\\\\right\\\\}\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[136,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[136,logseq____"^Flogseq____",125,536870920]],[logseq____"^15logseq____",[136,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[136,logseq____"^Vlogseq____",125,536870920]],[logseq____"^15logseq____",[136,logseq____"^Ulogseq____",111,536870920]],[logseq____"^15logseq____",[136,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[136,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[136,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[136,logseq____"^;logseq____",logseq____"~u652fe1d8-dd91-452c-9f3d-f41093f82bedlogseq____",536870920]],[logseq____"^15logseq____",[137,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 2}\\\\quad -5\\\\le xlogseq____<3$\\n\\\\begin{align}\\nx+5logseq____&\\\\le0\\\\\\\\\\n|x+5|logseq____&=x+5\\\\\\\\\\nx-3logseq____&logseq____<0\\\\\\\\\\n|x-3|logseq____&=-(x-3)=-x+3\\\\\\\\\\nx+5logseq____&logseq____<2(-x+3)\\\\\\\\\\nx+5logseq____&logseq____<-2x+6 logseq____&|logseq____&+2x\\\\\\\\\\n3xlogseq____&logseq____<1 logseq____&|logseq____& :3\\\\\\\\\\nxlogseq____&logseq____<\\\\frac13\\n\\\\end{align}\\n\\\\begin{align}\\nL_2=\\\\left\\\\{x\\\\in\\\\Reals\\\\middle|-5\\\\le xlogseq____<3,xlogseq____<\\\\frac13\\\\right\\\\}\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[137,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[137,logseq____"^Flogseq____",119,536870920]],[logseq____"^15logseq____",[137,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[137,logseq____"^Vlogseq____",140,536870920]],[logseq____"^15logseq____",[137,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[137,logseq____"^Ulogseq____",108,536870920]],[logseq____"^15logseq____",[137,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[137,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[137,logseq____"^Ulogseq____",115,536870920]],[logseq____"^15logseq____",[137,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[137,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[137,logseq____"^;logseq____",logseq____"~u652fe1d8-8864-4eb7-8f62-a8fb8a2ee5e4logseq____",536870920]],[logseq____"^15logseq____",[138,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 1}\\\\quad xlogseq____<3$\\n\\\\begin{align}\\n|x-3|logseq____&=-x+3\\\\\\\\\\n-x+3logseq____&logseq____<5 logseq____&|logseq____&-3\\\\\\\\\\n-xlogseq____&logseq____<2 logseq____&|logseq____&*-1\\\\\\\\\\nxlogseq____&logseq____>-2\\\\\\\\\\nL_1logseq____&=(-2,3)\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[138,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[138,logseq____"^Flogseq____",154,536870920]],[logseq____"^15logseq____",[138,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[138,logseq____"^Vlogseq____",154,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",108,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",115,536870920]],[logseq____"^15logseq____",[138,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[138,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[138,logseq____"^;logseq____",logseq____"~u652fe1d8-bf55-4510-8ae0-2077ef7cc39elogseq____",536870920]],[logseq____"^15logseq____",[139,logseq____"^Qlogseq____",logseq____"$\\\\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____& *(-1)\\\\\\\\\\nxlogseq____&logseq____>11\\n\\\\end{align}\\n\\\\begin{align}\\nL_3=\\\\left\\\\{x\\\\in\\\\Reals\\\\Big|xlogseq____>3,xlogseq____>11\\\\right\\\\}=(11,\\\\infty)\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[139,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[139,logseq____"^Flogseq____",137,536870920]],[logseq____"^15logseq____",[139,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[139,logseq____"^Vlogseq____",140,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",108,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",115,536870920]],[logseq____"^15logseq____",[139,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[139,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[139,logseq____"^;logseq____",logseq____"~u652fe1d8-8207-4c8d-9295-81c50ebdef1flogseq____",536870920]],[logseq____"^15logseq____",[140,logseq____"^Qlogseq____",logseq____"### Lösung mit [[Fallunterscheidung]]logseq____",536870920]],[logseq____"^15logseq____",[140,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[140,logseq____"^Flogseq____",128,536870920]],[logseq____"^15logseq____",[140,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[140,logseq____"^Vlogseq____",128,536870920]],[logseq____"^15logseq____",[140,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[140,logseq____"^Ulogseq____",108,536870920]],[logseq____"^15logseq____",[140,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[140,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[140,logseq____"^Ulogseq____",115,536870920]],[logseq____"^15logseq____",[140,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[140,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870920]],[logseq____"^15logseq____",[140,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[140,logseq____"^Hlogseq____",108,536870920]],[logseq____"^15logseq____",[140,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[140,logseq____"^;logseq____",logseq____"~u652fe1d8-589b-421e-85d8-193571fe41dclogseq____",536870920]],[logseq____"^15logseq____",[141,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 2b}\\\\quad xlogseq____<-2$\\n\\\\begin{align}\\n|\\\\underbrace{x+2}_{logseq____<0}logseq____&=-(x+2)=-x-2\\\\\\\\\\n-x+3logseq____&\\\\le -x-2 logseq____&|logseq____&+x\\\\\\\\\\n3logseq____&\\\\le-2logseq____&logseq____&\\\\underset{falsche Aussage}{f.A}\\\\\\\\\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[141,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[141,logseq____"^Flogseq____",156,536870920]],[logseq____"^15logseq____",[141,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[141,logseq____"^Vlogseq____",142,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",108,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",115,536870920]],[logseq____"^15logseq____",[141,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[141,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[141,logseq____"^;logseq____",logseq____"~u652fe1d8-4aa1-40ce-982d-665bc0fad071logseq____",536870920]],[logseq____"^15logseq____",[142,logseq____"^Qlogseq____",logseq____"### 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logseq____",logseq____"^18logseq____",3],536870920]],[logseq____"^15logseq____",[142,logseq____"^Jlogseq____",[],536870920]],[logseq____"^15logseq____",[142,logseq____"^Hlogseq____",108,536870920]],[logseq____"^15logseq____",[142,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[142,logseq____"^;logseq____",logseq____"~u652fe1d8-641b-4cd2-90dc-c30c57d14cealogseq____",536870920]],[logseq____"^15logseq____",[143,logseq____"^Qlogseq____",logseq____"logseq____",536870920]],[logseq____"^15logseq____",[143,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[143,logseq____"^Flogseq____",148,536870920]],[logseq____"^15logseq____",[143,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[143,logseq____"^Vlogseq____",157,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",115,536870920]],[logseq____"^15logseq____",[143,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[143,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[143,logseq____"^;logseq____",logseq____"~u652fe1d8-3ac7-4399-b5d2-7ef8b2897311logseq____",536870920]],[logseq____"^15logseq____",[144,logseq____"^Qlogseq____",logseq____"$\\\\text{Fall 2}\\\\quad xlogseq____<3$\\n\\\\begin{align}\\n|\\\\underbrace{x-3}_{logseq____<0}|logseq____&=6(x-3)=-x+3\\\\\\\\\\n-x+3logseq____&\\\\le|x+2|\\\\\\\\\\n\\\\end{align}logseq____",536870920]],[logseq____"^15logseq____",[144,logseq____"^Ologseq____",logseq____"^16logseq____",536870920]],[logseq____"^15logseq____",[144,logseq____"^Flogseq____",151,536870920]],[logseq____"^15logseq____",[144,logseq____"^Xlogseq____",114,536870920]],[logseq____"^15logseq____",[144,logseq____"^Vlogseq____",142,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",35,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",108,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",113,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",114,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",115,536870920]],[logseq____"^15logseq____",[144,logseq____"^Ulogseq____",118,536870920]],[logseq____"^15logseq____",[144,logseq____"^17logseq____",true,536870920]],[logseq____"^15logseq____",[144,logseq____"^;logseq____",logseq____"~u652fe1d8-fe41-4a7b-ac65-2fcf94f5ec4flogseq____",536870920]],[logseq____"^15logseq____",[145,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nM=\\\\left\\\\{\\\\underbrace{(x,y)}_\\\\text{Punkte 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logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[190,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[190,logseq____"^Hlogseq____",169,536870922]],[logseq____"^15logseq____",[190,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[190,logseq____"^;logseq____",logseq____"~u652fe1d8-cbb2-4aae-9c96-d1051512871flogseq____",536870922]],[logseq____"^15logseq____",[191,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870922]],[logseq____"^15logseq____",[191,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[191,logseq____"^Flogseq____",200,536870922]],[logseq____"^15logseq____",[191,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[191,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[191,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[191,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[191,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[191,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[191,logseq____"^Hlogseq____",35,536870922]],[logseq____"^15logseq____",[191,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[191,logseq____"^;logseq____",logseq____"~u652fe1d8-c56c-4065-9efd-48e5b142125clogseq____",536870922]],[logseq____"^15logseq____",[192,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____",[192,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[192,logseq____"^Flogseq____",196,536870922]],[logseq____"^15logseq____",[192,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[192,logseq____"^Vlogseq____",196,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[192,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[192,logseq____"^;logseq____",logseq____"~u652fe1d8-bd6f-4d0e-9079-c82b8534a0b5logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Qlogseq____",logseq____"C. Meinel, M. Mundhenk, [[\\logseq____"Mathematische Grundlagen der Informatik\\logseq____", 5. Auflage 2011]]logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Flogseq____",195,536870922]],[logseq____"^15logseq____",[193,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[193,logseq____"^Vlogseq____",195,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",173,536870922]],[logseq____"^15logseq____",[193,logseq____"^Hlogseq____",173,536870922]],[logseq____"^15logseq____",[193,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[193,logseq____"^;logseq____",logseq____"~u652fe1d8-9931-42a0-b034-0ca1f38ecb3alogseq____",536870922]],[logseq____"^15logseq____",[194,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____",[194,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[194,logseq____"^Flogseq____",191,536870922]],[logseq____"^15logseq____",[194,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[194,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",172,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",177,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",179,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",183,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[194,logseq____"^Hlogseq____",172,536870922]],[logseq____"^15logseq____",[194,logseq____"^Hlogseq____",177,536870922]],[logseq____"^15logseq____",[194,logseq____"^Hlogseq____",179,536870922]],[logseq____"^15logseq____",[194,logseq____"^Hlogseq____",183,536870922]],[logseq____"^15logseq____",[194,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[194,logseq____"^;logseq____",logseq____"~u652fe1d8-3171-456a-a4d2-78c47082e179logseq____",536870922]],[logseq____"^15logseq____",[195,logseq____"^Qlogseq____",logseq____"# Büchertiplogseq____",536870922]],[logseq____"^15logseq____",[195,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[195,logseq____"^Flogseq____",165,536870922]],[logseq____"^15logseq____",[195,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[195,logseq____"^Vlogseq____",165,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[195,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[195,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[195,logseq____"^17logseq____",false,536870922]],[logseq____"^15logseq____",[195,logseq____"^;logseq____",logseq____"~u652fe1d8-8f0b-4ef0-b71b-eff0bb1e5f9elogseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Qlogseq____",logseq____"Weitere [[Beispiele]]logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Flogseq____",198,536870922]],[logseq____"^15logseq____",[196,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[196,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[196,logseq____"^Hlogseq____",35,536870922]],[logseq____"^15logseq____",[196,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[196,logseq____"^;logseq____",logseq____"~u652fe1d8-4fcb-4a45-835d-3e78bb50883elogseq____",536870922]],[logseq____"^15logseq____",[197,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____",[197,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Flogseq____",203,536870922]],[logseq____"^15logseq____",[197,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[197,logseq____"^Vlogseq____",207,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[197,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[197,logseq____"^;logseq____",logseq____"~u652fe1d8-019c-42a5-be2d-c9425e24188elogseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Qlogseq____",logseq____"Alternativ: $p\\\\equiv q$, falls $p\\\\leftrightarrow q$ [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Flogseq____",210,536870922]],[logseq____"^15logseq____",[198,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[198,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[198,logseq____"^Hlogseq____",174,536870922]],[logseq____"^15logseq____",[198,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[198,logseq____"^;logseq____",logseq____"~u652fe1d8-393f-4153-a862-cc1f9e57b863logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Qlogseq____",logseq____"Formel, die konstant $f$ istlogseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Flogseq____",189,536870922]],[logseq____"^15logseq____",[199,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[199,logseq____"^Vlogseq____",189,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",175,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[199,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[199,logseq____"^;logseq____",logseq____"~u652fe1d8-e399-4712-881b-96222b7679aflogseq____",536870922]],[logseq____"^15logseq____",[200,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____",[200,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[200,logseq____"^Flogseq____",214,536870922]],[logseq____"^15logseq____",[200,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[200,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[200,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[200,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[200,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[200,logseq____"^Hlogseq____",180,536870922]],[logseq____"^15logseq____",[200,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[200,logseq____"^;logseq____",logseq____"~u652fe1d8-6d63-41f1-9447-4705c20f7c1blogseq____",536870922]],[logseq____"^15logseq____",[201,logseq____"^Qlogseq____",logseq____"Formel, die konstant $w$ istlogseq____",536870922]],[logseq____"^15logseq____",[201,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[201,logseq____"^Flogseq____",213,536870922]],[logseq____"^15logseq____",[201,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[201,logseq____"^Vlogseq____",213,536870922]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[201,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[201,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[201,logseq____"^;logseq____",logseq____"~u652fe1d8-bc05-4e45-b3d2-f57cdcdbfe55logseq____",536870922]],[logseq____"^15logseq____",[202,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\np\\\\land q\\\\lor r\\\\\\\\\\n(p\\\\land q)\\\\lor r\\\\\\\\\\np\\\\land (q\\\\lor r)\\\\\\\\\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[202,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[202,logseq____"^Flogseq____",191,536870922]],[logseq____"^15logseq____",[202,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[202,logseq____"^Vlogseq____",191,536870922]],[logseq____"^15logseq____",[202,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[202,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[202,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[202,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[202,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[202,logseq____"^;logseq____",logseq____"~u652fe1d8-0bf7-4f84-9a8c-1e4f3a5e02e1logseq____",536870922]],[logseq____"^15logseq____",[203,logseq____"^Qlogseq____",logseq____"Satz, der wahr oder falsch ist, d.h. der [[Wahrheitswert]] $w$ bzw $f$ hat ($t/f$, $1/0$, $\\\\text{wahr}/\\\\text{falsch}$)logseq____",536870922]],[logseq____"^15logseq____",[203,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[203,logseq____"^Flogseq____",207,536870922]],[logseq____"^15logseq____",[203,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[203,logseq____"^Vlogseq____",207,536870922]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",172,536870922]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[203,logseq____"^Hlogseq____",172,536870922]],[logseq____"^15logseq____",[203,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[203,logseq____"^;logseq____",logseq____"~u652fe1d8-e85b-4e77-b8a8-5e260bc41698logseq____",536870922]],[logseq____"^15logseq____",[204,logseq____"^Qlogseq____",logseq____"### [[Wahrheitswerteverlauf]] einer [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[204,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[204,logseq____"^Flogseq____",212,536870922]],[logseq____"^15logseq____",[204,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[204,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",182,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[204,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[204,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[204,logseq____"^Hlogseq____",168,536870922]],[logseq____"^15logseq____",[204,logseq____"^Hlogseq____",182,536870922]],[logseq____"^15logseq____",[204,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[204,logseq____"^;logseq____",logseq____"~u652fe1d8-c342-4ed4-b195-b8a551170469logseq____",536870922]],[logseq____"^15logseq____",[205,logseq____"^Qlogseq____",logseq____"Zuordnung von $w/f$ an jede [Variable]([[Variablen]]) einer [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[205,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[205,logseq____"^Flogseq____",212,536870922]],[logseq____"^15logseq____",[205,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[205,logseq____"^Vlogseq____",212,536870922]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",182,536870922]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",186,536870922]],[logseq____"^15logseq____",[205,logseq____"^Hlogseq____",176,536870922]],[logseq____"^15logseq____",[205,logseq____"^Hlogseq____",182,536870922]],[logseq____"^15logseq____",[205,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[205,logseq____"^;logseq____",logseq____"~u652fe1d8-d89a-4baf-8d06-cfcb742e3769logseq____",536870922]],[logseq____"^15logseq____",[206,logseq____"^Qlogseq____",logseq____"$\\\\text{Beispiel: Bestimmung des }$[[Wahrheitswerteverlauf]]$\\\\text{ von }(p\\\\rightarrow q)\\\\land(q\\\\rightarrow p)$\\n|$p$|$q$||$p\\\\rightarrow q$|$q\\\\rightarrow p$|$(p\\\\rightarrow q)\\\\land(q\\\\rightarrow p)|$p\\\\leftrightarrow q$|\\n|-|-|-|-|-|-|\\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____",[206,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[206,logseq____"^Flogseq____",211,536870922]],[logseq____"^15logseq____",[206,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[206,logseq____"^Vlogseq____",204,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",182,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[206,logseq____"^Hlogseq____",168,536870922]],[logseq____"^15logseq____",[206,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[206,logseq____"^;logseq____",logseq____"~u652fe1d8-1e63-468e-904d-51f4f09136f2logseq____",536870922]],[logseq____"^15logseq____",[207,logseq____"^Qlogseq____",logseq____"# 1. [[Aussagen]]logseq____",536870922]],[logseq____"^15logseq____",[207,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[207,logseq____"^Flogseq____",195,536870922]],[logseq____"^15logseq____",[207,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[207,logseq____"^Vlogseq____",165,536870922]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[207,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[207,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[207,logseq____"^Hlogseq____",180,536870922]],[logseq____"^15logseq____",[207,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[207,logseq____"^;logseq____",logseq____"~u652fe1d8-5f98-4d25-99c5-fcf4636975c3logseq____",536870922]],[logseq____"^15logseq____",[208,logseq____"^Qlogseq____",logseq____"Entsteht durch sukzessive [[Verknüpfungen]] wie oben an Variablen ergibt #FIXMElogseq____",536870922]],[logseq____"^15logseq____",[208,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[208,logseq____"^Flogseq____",187,536870922]],[logseq____"^15logseq____",[208,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[208,logseq____"^Vlogseq____",187,536870922]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",42,536870922]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",181,536870922]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",182,536870922]],[logseq____"^15logseq____",[208,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[208,logseq____"^Hlogseq____",42,536870922]],[logseq____"^15logseq____",[208,logseq____"^Hlogseq____",181,536870922]],[logseq____"^15logseq____",[208,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[208,logseq____"^;logseq____",logseq____"~u652fe1d8-6fb9-4eeb-b64a-48ef7cae4d24logseq____",536870922]],[logseq____"^15logseq____",[209,logseq____"^Qlogseq____",logseq____"[Variable]([[Variablen]]), die den Wert $w$ oder $f$ annimmtlogseq____",536870922]],[logseq____"^15logseq____",[209,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[209,logseq____"^Flogseq____",190,536870922]],[logseq____"^15logseq____",[209,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[209,logseq____"^Vlogseq____",190,536870922]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",169,536870922]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[209,logseq____"^Hlogseq____",176,536870922]],[logseq____"^15logseq____",[209,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[209,logseq____"^;logseq____",logseq____"~u652fe1d8-6b5e-40ac-99dd-c0b6c2f17c0flogseq____",536870922]],[logseq____"^15logseq____",[210,logseq____"^Qlogseq____",logseq____"Formeln $p,q$ [logisch äquivalent]([[Logisch Äquivalent]]), wenn sie den gleichen [[Wahrheitswerteverlauf]] haben.\\n[[Schreibweise]]: $p\\\\equiv q$logseq____",536870922]],[logseq____"^15logseq____",[210,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[210,logseq____"^Flogseq____",189,536870922]],[logseq____"^15logseq____",[210,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[210,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",185,536870922]],[logseq____"^15logseq____",[210,logseq____"^Hlogseq____",168,536870922]],[logseq____"^15logseq____",[210,logseq____"^Hlogseq____",178,536870922]],[logseq____"^15logseq____",[210,logseq____"^Hlogseq____",185,536870922]],[logseq____"^15logseq____",[210,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[210,logseq____"^;logseq____",logseq____"~u652fe1d8-f8a6-4e3b-9f00-b2fff017ef02logseq____",536870922]],[logseq____"^15logseq____",[211,logseq____"^Qlogseq____",logseq____"Jede mögliche [[Belegung]] wird ein [[Wahrheitswert]] der [[Formel]] zugeordnet (nach [[Tabellenregeln]])logseq____",536870922]],[logseq____"^15logseq____",[211,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[211,logseq____"^Flogseq____",204,536870922]],[logseq____"^15logseq____",[211,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[211,logseq____"^Vlogseq____",204,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",171,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",172,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",179,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",182,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",186,536870922]],[logseq____"^15logseq____",[211,logseq____"^Hlogseq____",171,536870922]],[logseq____"^15logseq____",[211,logseq____"^Hlogseq____",172,536870922]],[logseq____"^15logseq____",[211,logseq____"^Hlogseq____",179,536870922]],[logseq____"^15logseq____",[211,logseq____"^Hlogseq____",186,536870922]],[logseq____"^15logseq____",[211,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[211,logseq____"^;logseq____",logseq____"~u652fe1d8-d5d5-4d1b-9588-b9e749a6c4b0logseq____",536870922]],[logseq____"^15logseq____",[212,logseq____"^Qlogseq____",logseq____"### [[Belegung]] (der [[Variablen]])logseq____",536870922]],[logseq____"^15logseq____",[212,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[212,logseq____"^Flogseq____",187,536870922]],[logseq____"^15logseq____",[212,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[212,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",186,536870922]],[logseq____"^15logseq____",[212,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[212,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[212,logseq____"^Hlogseq____",176,536870922]],[logseq____"^15logseq____",[212,logseq____"^Hlogseq____",186,536870922]],[logseq____"^15logseq____",[212,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[212,logseq____"^;logseq____",logseq____"~u652fe1d8-e28a-490b-befa-a717b6be1703logseq____",536870922]],[logseq____"^15logseq____",[213,logseq____"^Qlogseq____",logseq____"### [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[213,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[213,logseq____"^Flogseq____",204,536870922]],[logseq____"^15logseq____",[213,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[213,logseq____"^Vlogseq____",214,536870922]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[213,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[213,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[213,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[213,logseq____"^Hlogseq____",174,536870922]],[logseq____"^15logseq____",[213,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[213,logseq____"^;logseq____",logseq____"~u652fe1d8-f1a6-4880-8adb-2dd6e703bc83logseq____",536870922]],[logseq____"^15logseq____",[214,logseq____"^Qlogseq____",logseq____"## [[Verknüpfung von 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morgansche regelnlogseq____",536870923]],[logseq____"^15logseq____",[217,logseq____"^11logseq____",logseq____"de Morgansche 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logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[246,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[246,logseq____"^Hlogseq____",216,536870923]],[logseq____"^15logseq____",[246,logseq____"^17logseq____",false,536870923]],[logseq____"^15logseq____",[246,logseq____"^;logseq____",logseq____"~u652fe1d9-e1bd-400a-af99-8c4d193ffabflogseq____",536870923]],[logseq____"^15logseq____",[247,logseq____"^Qlogseq____",logseq____"Sei $p(x,y)$ AF in Variablen $x,y$ über $U_1,U_2$\\n\\\\begin{align}\\n\\\\underbrace{\\\\forall x: \\\\underbrace{\\\\exists y: p(x,y)}_{\\\\text{AF in den Variablen }x\\\\text{ über }U_1}}_{\\\\text{Aussage}}\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[247,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[247,logseq____"^Flogseq____",264,536870923]],[logseq____"^15logseq____",[247,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[247,logseq____"^Vlogseq____",264,536870923]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",216,536870923]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[247,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[247,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[247,logseq____"^;logseq____",logseq____"~u652fe1d9-c39e-4e9d-809c-80cebb41d7a2logseq____",536870923]],[logseq____"^15logseq____",[248,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____",[248,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[248,logseq____"^Flogseq____",274,536870923]],[logseq____"^15logseq____",[248,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[248,logseq____"^Vlogseq____",274,536870923]],[logseq____"^15logseq____",[248,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[248,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[248,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[248,logseq____"^;logseq____",logseq____"~u652fe1d9-b54c-4676-b00d-27aaebad1bf9logseq____",536870923]],[logseq____"^15logseq____",[249,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nlogseq____&\\\\forall x: \\\\forall y: p(x, y) \\\\equiv \\\\forall y: \\\\forall x: p(x,y)\\\\\\\\\\nlogseq____&\\\\exists x:\\\\exists y:p(x,y)\\\\equiv\\\\exists y:\\\\exists x:p(x,y)\\\\\\\\\\nlogseq____&[\\\\forall x_1:\\\\forall x_2: \\\\cdots:\\\\forall x_n:p(x_1,x_2,\\\\cdots,x_n)\\\\notag\\\\\\\\\\nlogseq____&\\\\equiv\\\\forall_{x_{\\\\pi(1)}}:\\\\forall_{x_{\\\\pi(1)}}:\\\\cdots:\\\\forall_{x_{\\\\pi(1)}}:p(x_1,x_2,\\\\cdots,x_n)\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[249,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[249,logseq____"^Flogseq____",256,536870923]],[logseq____"^15logseq____",[249,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[249,logseq____"^Vlogseq____",256,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[249,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[249,logseq____"^;logseq____",logseq____"~u652fe1d9-43b0-4143-96ef-c99f45f3e453logseq____",536870923]],[logseq____"^15logseq____",[250,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:(p(x)\\\\land q(x)) logseq____&\\\\equiv (\\\\forall x: p(x)) \\\\land (\\\\forall x:q(x));\\\\notag\\\\\\\\\\n\\\\exists x: (p(x) \\\\lor q(x)) logseq____&\\\\equiv (\\\\exists x: p(x))\\\\lor(\\\\exists x: q(x)),\\\\quadlogseq____&\\\\text{Große Assoziativgesetze}\\\\\\\\\\n\\\\neg(\\\\forall x:p(x))logseq____&\\\\equiv \\\\exists x: (\\\\neg p(x));\\\\quadlogseq____&\\\\text{Große de Morgansche Regeln;}\\\\notag\\\\\\\\\\n\\\\neg(\\\\exists x: p(x)) logseq____&\\\\equiv \\\\forall x: (\\\\neg p(x)),\\\\quadlogseq____&\\\\text{Negationsregeln.}\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[250,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[250,logseq____"^Flogseq____",277,536870923]],[logseq____"^15logseq____",[250,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[250,logseq____"^Vlogseq____",277,536870923]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",216,536870923]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[250,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[250,logseq____"^;logseq____",logseq____"~u652fe1d9-e0a2-4733-afe2-c64525a0bfe3logseq____",536870923]],[logseq____"^15logseq____",[251,logseq____"^Qlogseq____",logseq____"Alternativschreibweisen:\\n$\\\\forall x: p(x)$ auch $\\\\bigwedge x:p(x)$\\n$\\\\exists x:p(x)$ auch $\\\\bigvee x:p(x)$logseq____",536870923]],[logseq____"^15logseq____",[251,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[251,logseq____"^Flogseq____",241,536870923]],[logseq____"^15logseq____",[251,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[251,logseq____"^Vlogseq____",265,536870923]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[251,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[251,logseq____"^;logseq____",logseq____"~u652fe1d9-fc8e-4d63-b48f-b4b0556b6ca7logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Flogseq____",249,536870923]],[logseq____"^15logseq____",[252,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[252,logseq____"^Vlogseq____",256,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[252,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[252,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[252,logseq____"^;logseq____",logseq____"~u652fe1d9-64f5-4e6e-87e5-0fc74b697f77logseq____",536870923]],[logseq____"^15logseq____",[253,logseq____"^Qlogseq____",logseq____"#FIXMElogseq____",536870923]],[logseq____"^15logseq____",[253,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[253,logseq____"^Flogseq____",251,536870923]],[logseq____"^15logseq____",[253,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[253,logseq____"^Vlogseq____",251,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",42,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[253,logseq____"^Hlogseq____",42,536870923]],[logseq____"^15logseq____",[253,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[253,logseq____"^;logseq____",logseq____"~u652fe1d9-5851-4f5b-90f2-08bcef64eddelogseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nlogseq____&\\\\neg\\\\left(\\\\forall x :\\\\exists y :\\\\forall w :\\\\forall u :\\\\exists z:p(x,y,z)\\\\rightarrow q(u,w)\\\\right)\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:(\\\\neg(\\\\exists y: \\\\forall w: \\\\forall u: \\\\exists z: p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:\\\\forall y:(\\\\neg(\\\\forall w: \\\\forall u:\\\\exists z: p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x:\\\\forall y:\\\\exists w:(\\\\neg(\\\\forall u:\\\\exists z:p(x,y,z)\\\\rightarrow q(u,w)))\\\\\\\\\\n\\\\equivlogseq____&\\\\exists x: \\\\forall y: \\\\exists w: \\\\exists u : \\\\forall z:\\\\neg(p(x,y,z)\\\\rightarrow q(u,w))\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Flogseq____",273,536870923]],[logseq____"^15logseq____",[254,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[254,logseq____"^Vlogseq____",273,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[254,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[254,logseq____"^;logseq____",logseq____"~u652fe1d8-b87c-4651-b28d-f491ae06c054logseq____",536870923]],[logseq____"^15logseq____",[255,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____",[255,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[255,logseq____"^Flogseq____",269,536870923]],[logseq____"^15logseq____",[255,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[255,logseq____"^Vlogseq____",265,536870923]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[255,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[255,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[255,logseq____"^;logseq____",logseq____"~u652fe1d9-aab4-495e-b9d7-c97ce8c2cea6logseq____",536870923]],[logseq____"^15logseq____",[256,logseq____"^Qlogseq____",logseq____"[Gleichartige]([[Gleichartig]]) [[Quantoren]] dürfen vertauscht werdenlogseq____",536870923]],[logseq____"^15logseq____",[256,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[256,logseq____"^Flogseq____",277,536870923]],[logseq____"^15logseq____",[256,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[256,logseq____"^Vlogseq____",265,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[256,logseq____"^Hlogseq____",224,536870923]],[logseq____"^15logseq____",[256,logseq____"^Hlogseq____",228,536870923]],[logseq____"^15logseq____",[256,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[256,logseq____"^;logseq____",logseq____"~u652fe1d9-7a6e-4357-aa04-9d206bdca0d7logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Qlogseq____",logseq____"Sei $x$ aus $\\\\Reals$ beliebig. Setze $y:=x+1$ (ist aus $\\\\Reals$). Dann ist $xlogseq____<x+1=y$ also $xlogseq____<y$. $\\\\blacksquare$logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Flogseq____",243,536870923]],[logseq____"^15logseq____",[257,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[257,logseq____"^Vlogseq____",243,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",227,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[257,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[257,logseq____"^;logseq____",logseq____"~u652fe1d8-416c-45b7-b321-82b02f61da10logseq____",536870923]],[logseq____"^15logseq____",[258,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____",[258,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Flogseq____",278,536870923]],[logseq____"^15logseq____",[258,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[258,logseq____"^Vlogseq____",265,536870923]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[258,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[258,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[258,logseq____"^;logseq____",logseq____"~u652fe1d9-3738-4520-b398-d8f4017cb774logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Qlogseq____",logseq____"$\\\\forall x:p(x)$ ist die [Aussage]([[Aussagen]]): $\\logseq____"\\\\text{Für alle x aus U ist p(x) wahr}\\logseq____"\\nIhr [[Wahrheitswert]] ist $\\\\begin{cases}w\\\\quadlogseq____&\\\\text{fallls p(x) w für alle x aus U}\\\\\\\\f\\\\quadlogseq____&\\\\text{falls p(x) f für (wenigstens) ein x aus U}\\\\end{cases}$logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Flogseq____",272,536870923]],[logseq____"^15logseq____",[259,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[259,logseq____"^Vlogseq____",272,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",172,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",180,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[259,logseq____"^Hlogseq____",172,536870923]],[logseq____"^15logseq____",[259,logseq____"^Hlogseq____",180,536870923]],[logseq____"^15logseq____",[259,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[259,logseq____"^;logseq____",logseq____"~u652fe1d9-f4a6-4c93-8689-22ebabcb33belogseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Qlogseq____",logseq____"see: [[Allquantor]] [[Existenzquantor]]logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Flogseq____",258,536870923]],[logseq____"^15logseq____",[260,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[260,logseq____"^Vlogseq____",258,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",229,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",235,536870923]],[logseq____"^15logseq____",[260,logseq____"^Hlogseq____",229,536870923]],[logseq____"^15logseq____",[260,logseq____"^Hlogseq____",235,536870923]],[logseq____"^15logseq____",[260,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[260,logseq____"^;logseq____",logseq____"~u652fe1d9-64ab-4bda-af35-c90166e77774logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist 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[[Quantoren]] dürfen **nicht** vertauscht werdenlogseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Flogseq____",242,536870923]],[logseq____"^15logseq____",[262,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[262,logseq____"^Vlogseq____",249,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[262,logseq____"^Hlogseq____",228,536870923]],[logseq____"^15logseq____",[262,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[262,logseq____"^;logseq____",logseq____"~u652fe1d9-cdfa-4fe0-9de1-4f76d182733dlogseq____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Flogseq____",268,536870923]],[logseq____"^15logseq____",[263,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[263,logseq____"^Vlogseq____",265,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[263,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[263,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[263,logseq____"^;logseq____",logseq____"~u652fe1d9-ba1c-440e-b74d-274f9e49c0f4logseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Qlogseq____",logseq____"Schachtelung von quantoren ([Beispiel]([[Beispiele]])):logseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Flogseq____",250,536870923]],[logseq____"^15logseq____",[264,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[264,logseq____"^Vlogseq____",277,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",216,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[264,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[264,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[264,logseq____"^;logseq____",logseq____"~u652fe1d9-ae9a-40eb-9e17-91492d005467logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Qlogseq____",logseq____"# [[Aussageform]]logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Flogseq____",270,536870923]],[logseq____"^15logseq____",[265,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[265,logseq____"^Vlogseq____",223,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[265,logseq____"^?logseq____",[logseq____"^ 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(AF) über [Universen]([[Universen (Logik)]]) $U_1,\\\\cdots,U_n$:logseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Flogseq____",265,536870923]],[logseq____"^15logseq____",[267,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[267,logseq____"^Vlogseq____",265,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",238,536870923]],[logseq____"^15logseq____",[267,logseq____"^Hlogseq____",234,536870923]],[logseq____"^15logseq____",[267,logseq____"^Hlogseq____",238,536870923]],[logseq____"^15logseq____",[267,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[267,logseq____"^;logseq____",logseq____"~u652fe1d9-1f8b-47ba-96c8-ea4fa014161elogseq____",536870923]],[logseq____"^15logseq____",[268,logseq____"^Qlogseq____",logseq____"[[Satz]] mit [[Variablen]] $x_1,\\\\cdots,x_n$, der bei Ersetzung jedes $x_j$ durch ein [Objekt]([[Objekt (Logik)]]) aus $U_j$ stets zu einer [Aussage]([[Aussagen]]) 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beiden Quantoren $\\\\forall, \\\\exists$ erweitern die [Sprache]([[Sprachen]]) der 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in der Variablen $x$ über 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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 Aussagen}\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[274,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[274,logseq____"^Flogseq____",263,536870923]],[logseq____"^15logseq____",[274,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[274,logseq____"^Vlogseq____",265,536870923]],[logseq____"^15logseq____",[274,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[274,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[274,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[274,logseq____"^;logseq____",logseq____"~u652fe1d9-1e4d-4898-a42d-cd381499e0c9logseq____",536870923]],[logseq____"^15logseq____",[275,logseq____"^Qlogseq____",logseq____"see: [[assoziativ]] [[kommutativ]] [[idenpotent]] [[wechselseitig distributiv]] [[distributiv]] [[de Morgansche Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[275,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[275,logseq____"^Flogseq____",270,536870923]],[logseq____"^15logseq____",[275,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[275,logseq____"^Vlogseq____",270,536870923]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",225,536870923]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",226,536870923]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",232,536870923]],[logseq____"^15logseq____",[275,logseq____"^Ulogseq____",233,536870923]],[logseq____"^15logseq____",[275,logseq____"^Hlogseq____",217,536870923]],[logseq____"^15logseq____",[275,logseq____"^Hlogseq____",222,536870923]],[logseq____"^15logseq____",[275,logseq____"^Hlogseq____",225,536870923]],[logseq____"^15logseq____",[275,logseq____"^Hlogseq____",226,536870923]],[logseq____"^15logseq____",[275,logseq____"^Hlogseq____",232,536870923]],[logseq____"^15logseq____",[275,logseq____"^Hlogseq____",233,536870923]],[logseq____"^15logseq____",[275,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[275,logseq____"^;logseq____",logseq____"~u652fe1d9-8b4a-44f0-8e8a-b99ed7c2e014logseq____",536870923]],[logseq____"^15logseq____",[276,logseq____"^Qlogseq____",logseq____"$\\\\blacksquare$ = Beweisende, auch [q.e.d.]([[Q.E.D.]]), [[w.z.b.w.]], QED, 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