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\\\\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____",[54,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Flogseq____",86,536870919]],[logseq____"^15logseq____",[54,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[54,logseq____"^;logseq____",logseq____"~u652bbc48-be49-4031-a935-8b5040312e60logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(2x+y)^4logseq____&=(2x)^4+4*(2x)^3y+6*(2x)^2y^2+4*2xy^3+y^4\\\\notag\\\\\\\\\\nlogseq____&=16x^4+32x^3y+24x^2y^2+8xy^3+y^4\\\\\\\\\\n(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5logseq____&=2+20x+40x^2\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Flogseq____",88,536870919]],[logseq____"^15logseq____",[55,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^Vlogseq____",88,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[55,logseq____"^;logseq____",logseq____"~u652bbc48-3d55-4359-ac4a-054ae06ee84clogseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Qlogseq____",logseq____"## [[Rationalmachen]] des [[Nenner]]slogseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Flogseq____",53,536870919]],[logseq____"^15logseq____",[56,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Vlogseq____",83,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[56,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[56,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[56,logseq____"^Hlogseq____",39,536870919]],[logseq____"^15logseq____",[56,logseq____"^Hlogseq____",45,536870919]],[logseq____"^15logseq____",[56,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[56,logseq____"^;logseq____",logseq____"~u652bbc48-6fa1-4348-ab11-1ddb5300c68clogseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Qlogseq____",logseq____"[[Wolframalpha]]logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Flogseq____",76,536870919]],[logseq____"^15logseq____",[57,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[57,logseq____"^Hlogseq____",46,536870919]],[logseq____"^15logseq____",[57,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[57,logseq____"^;logseq____",logseq____"~u652bbc48-acc8-4908-8d5f-a47248c2169blogseq____",536870919]],[logseq____"^15logseq____",[58,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____",[58,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Flogseq____",83,536870919]],[logseq____"^15logseq____",[58,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^Vlogseq____",83,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[58,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[58,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[58,logseq____"^;logseq____",logseq____"~u652bbc48-941b-4ae9-8b9e-327b52dde9b0logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Qlogseq____",logseq____"logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Flogseq____",66,536870919]],[logseq____"^15logseq____",[59,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Vlogseq____",90,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[59,logseq____"^;logseq____",logseq____"~u652bbc48-b53c-4e5f-97d5-476ef3a49b14logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Qlogseq____",logseq____"$\\\\text{Häufige Falle:}$\\n\\\\begin{align}\\n\\\\left((-1)^2\\\\right)^\\\\frac12logseq____&=(-1)^\\\\frac22logseq____&=(-1)^1logseq____&=-1\\\\\\\\\\n\\\\left((-1)^2\\\\right)^\\\\frac12logseq____&=1^\\\\frac12logseq____&logseq____&=1\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Flogseq____",92,536870919]],[logseq____"^15logseq____",[60,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Vlogseq____",67,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[60,logseq____"^;logseq____",logseq____"~u652bbc48-5460-4289-b28c-d553cc7c4b00logseq____",536870919]],[logseq____"^15logseq____",[61,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____",[61,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Flogseq____",56,536870919]],[logseq____"^15logseq____",[61,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^Vlogseq____",56,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[61,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[61,logseq____"^;logseq____",logseq____"~u652bbc48-1737-434b-8958-6c61f1f23d39logseq____",536870919]],[logseq____"^15logseq____",[62,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____",[62,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Flogseq____",97,536870919]],[logseq____"^15logseq____",[62,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^Vlogseq____",64,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[62,logseq____"^;logseq____",logseq____"~u652bbc48-5e91-443b-bce7-47c576f50a46logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Qlogseq____",logseq____"## [[pq-Formel]]logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Flogseq____",104,536870919]],[logseq____"^15logseq____",[63,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^Vlogseq____",79,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",29,536870919]],[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____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[63,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[63,logseq____"^Hlogseq____",43,536870919]],[logseq____"^15logseq____",[63,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[63,logseq____"^;logseq____",logseq____"~u652bbc48-91ce-40dc-8728-eca0c9e36370logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Qlogseq____",logseq____"# [[Binomische Formeln]]logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Flogseq____",71,536870919]],[logseq____"^15logseq____",[64,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[64,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[64,logseq____"^Hlogseq____",34,536870919]],[logseq____"^15logseq____",[64,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[64,logseq____"^;logseq____",logseq____"~u652bbc48-e584-4da9-90b1-357cbc930506logseq____",536870919]],[logseq____"^15logseq____",[65,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____",[65,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Flogseq____",99,536870919]],[logseq____"^15logseq____",[65,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Vlogseq____",63,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[65,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[65,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[65,logseq____"^;logseq____",logseq____"~u652bbc48-8da3-4929-b133-390473c3e49clogseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche }$[[Gleichungen]]$\\\\text{ sind richtig?}$\\n\\\\begin{align}\\ne^{x+y} logseq____&= e^x + e^y logseq____& ,f \\\\\\\\\\ne^{x+y} logseq____&= e^x * x^y logseq____& ,r \\\\\\\\\\ne^{x+y}logseq____&=e^{xy} logseq____& ,f\\\\\\\\\\ne^{ln(x+y)}logseq____&=x+y logseq____& ,x+ylogseq____>c \\\\\\\\\\ne^{ln(x+y)}logseq____&=e^x*e^ylogseq____&,f\\\\\\\\\\ne^{ln(x+y)}logseq____&=ln(e^x+y)logseq____&,x+ylogseq____>0\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Flogseq____",90,536870919]],[logseq____"^15logseq____",[66,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^Vlogseq____",90,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",33,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^Hlogseq____",33,536870919]],[logseq____"^15logseq____",[66,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[66,logseq____"^;logseq____",logseq____"~u652bbc48-8fef-4525-979b-9df8d8403fe6logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Flogseq____",49,536870919]],[logseq____"^15logseq____",[67,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[67,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[67,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[67,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[67,logseq____"^;logseq____",logseq____"~u652bbc48-382c-4188-9fa6-8b3e4044bad8logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Qlogseq____",logseq____"$\\\\frac12 \\\\sqrt2, \\\\frac12, \\\\frac12\\\\sqrt3 \\\\text{ sind werte von }\\\\sin x$\\n\\\\begin{align}\\n\\\\sin 30\\\\degreelogseq____&=\\\\frac12\\\\\\\\\\n\\\\sin 45\\\\degree logseq____&= \\\\frac12\\\\sqrt2 \\\\\\\\\\n\\\\sin 60\\\\degree logseq____&= \\\\frac12 \\\\sqrt3 \\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Flogseq____",71,536870919]],[logseq____"^15logseq____",[68,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Vlogseq____",71,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[68,logseq____"^;logseq____",logseq____"~u652bbc48-d835-43fc-8344-a44c23e47465logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Qlogseq____",logseq____"\\\\begin{array}{c} (a+b)^0logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&1 \\\\\\\\\\n(a+b)^1logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&logseq____&1 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^2logseq____&logseq____&logseq____&logseq____&logseq____&logseq____& 1 logseq____&logseq____& 2 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^3logseq____&logseq____&logseq____&logseq____&logseq____&1 logseq____&logseq____& 3 logseq____&logseq____& 3 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^4logseq____&logseq____&logseq____&logseq____& 1 logseq____&logseq____& 4 logseq____&logseq____& 6 logseq____&logseq____& 4 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^5logseq____&logseq____&logseq____& 1 logseq____&logseq____& 5 logseq____&logseq____& 10 logseq____&logseq____& 10 logseq____&logseq____& 5 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^6logseq____&logseq____& 1 logseq____&logseq____& 6 logseq____&logseq____& 15 logseq____&logseq____& 20 logseq____&logseq____& 15 logseq____&logseq____& 6 logseq____&logseq____& 1 \\\\\\\\\\n(a+b)^7logseq____&1 logseq____&logseq____& 7 logseq____&logseq____&21 logseq____&logseq____& 35 logseq____&logseq____& 35 logseq____&logseq____& \\\\cdots\\\\end{array}logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Flogseq____",48,536870919]],[logseq____"^15logseq____",[69,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^Vlogseq____",48,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[69,logseq____"^;logseq____",logseq____"~u652bbc48-db17-4305-ae86-c310e01e5444logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Qlogseq____",logseq____"#FIXME $\\\\sin^{-2}x+cos^2y=1,r$ ist falschlogseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Flogseq____",84,536870919]],[logseq____"^15logseq____",[70,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^Vlogseq____",84,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[70,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[70,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[70,logseq____"^;logseq____",logseq____"~u652bbc48-95c5-4ea7-80ee-8164c9f38b74logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Trigonometrie]]logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Flogseq____",82,536870919]],[logseq____"^15logseq____",[71,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[71,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[71,logseq____"^Hlogseq____",36,536870919]],[logseq____"^15logseq____",[71,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[71,logseq____"^;logseq____",logseq____"~u652bbc48-80d6-4ba4-ad63-756b0d4b7719logseq____",536870919]],[logseq____"^15logseq____",[72,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____",[72,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Flogseq____",73,536870919]],[logseq____"^15logseq____",[72,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Vlogseq____",73,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[72,logseq____"^;logseq____",logseq____"~u652bbc48-56f3-4303-90fb-10c6ec8170a8logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Qlogseq____",logseq____"# [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Flogseq____",91,536870919]],[logseq____"^15logseq____",[73,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^Vlogseq____",91,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[73,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[73,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[73,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[73,logseq____"^;logseq____",logseq____"~u652bbc48-57db-4ceb-a708-0cc60d53507elogseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Qlogseq____",logseq____"## [[Binomialkoeffizient]]logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Flogseq____",88,536870919]],[logseq____"^15logseq____",[74,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^Vlogseq____",48,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[74,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[74,logseq____"^Hlogseq____",32,536870919]],[logseq____"^15logseq____",[74,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[74,logseq____"^;logseq____",logseq____"~u652bbc48-8ca3-4540-95da-49d53c181917logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Flogseq____",72,536870919]],[logseq____"^15logseq____",[75,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^Vlogseq____",73,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[75,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[75,logseq____"^;logseq____",logseq____"~u652bbc48-4c19-4549-966e-ca5cb5a6e7a7logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Qlogseq____",logseq____"# [[Links]]logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Flogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[76,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[76,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[76,logseq____"^Hlogseq____",44,536870919]],[logseq____"^15logseq____",[76,logseq____"^17logseq____",false,536870919]],[logseq____"^15logseq____",[76,logseq____"^;logseq____",logseq____"~u652bbc48-9e63-4463-b186-dc5428abebaclogseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Qlogseq____",logseq____"$\\\\text{Herleitung mit }$[[Quadratische Ergänzung]]\\n\\\\begin{align}\\nx^2+px+qlogseq____&=0logseq____&|logseq____&-q\\\\\\\\\\nx^2+pxlogseq____&=-qlogseq____&|logseq____&+\\\\left(\\\\frac p2\\\\right)^2\\\\\\\\\\nx^2+px+\\\\left(\\\\frac p2\\\\right)^2logseq____&=-q+\\\\left(\\\\frac p2\\\\right)^2\\\\\\\\\\n\\\\underbrace{\\\\left(x+\\\\frac p2\\\\right)^2}_\\\\text{1. binomische Formel}logseq____&=-q+\\\\left(\\\\frac p2\\\\right)^2logseq____&|logseq____&\\\\sqrt{()}\\\\\\\\\\n\\\\left|x+\\\\frac p2\\\\right|logseq____&=\\\\sqrt{\\\\frac{p^2}4-q}logseq____&,logseq____&\\\\text{ falls }\\\\frac{p^2}4-q\\\\ge0\\\\\\\\\\nx+\\\\frac p2logseq____&=\\\\pm\\\\sqrt{\\\\frac{p^2}4-q}logseq____&|logseq____&-\\\\frac p2\\\\\\\\\\nxlogseq____&=-\\\\frac p2\\\\pm\\\\sqrt{\\\\frac{p^2}4-q}\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Flogseq____",63,536870919]],[logseq____"^15logseq____",[77,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^Vlogseq____",63,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",26,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[77,logseq____"^Hlogseq____",26,536870919]],[logseq____"^15logseq____",[77,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[77,logseq____"^;logseq____",logseq____"~u652bbc48-3e54-4791-82ff-f99638cacdc3logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Qlogseq____",logseq____"\\\\begin{array}{c}\\n{0\\\\choose 0}\\n\\\\\\\\\\n{1\\\\choose 0} {1\\\\choose 1}\\n\\\\\\\\\\n{2\\\\choose 0} {2\\\\choose 1} {2\\\\choose 2}\\n\\\\\\\\\\n{3\\\\choose 0} {3\\\\choose 1} {3\\\\choose 2} {3\\\\choose 3}\\n\\\\\\\\\\n\\\\cdots\\n\\\\end{array}logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Flogseq____",98,536870919]],[logseq____"^15logseq____",[78,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Vlogseq____",74,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[78,logseq____"^;logseq____",logseq____"~u652bbc48-e9bc-4337-a3e2-157c7470af70logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Qlogseq____",logseq____"# [[Quadratische Gleichungen]]logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Flogseq____",83,536870919]],[logseq____"^15logseq____",[79,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Vlogseq____",91,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[79,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[79,logseq____"^Hlogseq____",29,536870919]],[logseq____"^15logseq____",[79,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[79,logseq____"^;logseq____",logseq____"~u652bbc48-0b97-4fca-8df7-b4523cd53486logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Flogseq____",99,536870919]],[logseq____"^15logseq____",[80,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^Vlogseq____",99,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[80,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[80,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[80,logseq____"^;logseq____",logseq____"~u652bbc48-e2de-4f07-a54f-a51c90efa5c3logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Qlogseq____",logseq____"# [[Wurzelgleichung]]logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Flogseq____",79,536870919]],[logseq____"^15logseq____",[81,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^Vlogseq____",91,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[81,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[81,logseq____"^Hlogseq____",30,536870919]],[logseq____"^15logseq____",[81,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[81,logseq____"^;logseq____",logseq____"~u652bbc48-60cd-41cb-b5d8-b5771e0c9786logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Qlogseq____",logseq____"# Was ist Größer?logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Flogseq____",67,536870919]],[logseq____"^15logseq____",[82,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[82,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[82,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[82,logseq____"^;logseq____",logseq____"~u652bbc48-edc9-44c1-8644-2c6058ee71c7logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Qlogseq____",logseq____"# [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Flogseq____",73,536870919]],[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____",37,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[83,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[83,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[83,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[83,logseq____"^;logseq____",logseq____"~u652bbc48-b52b-4ec4-8e93-96f38278fca8logseq____",536870919]],[logseq____"^15logseq____",[84,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____",[84,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Flogseq____",68,536870919]],[logseq____"^15logseq____",[84,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^Vlogseq____",71,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[84,logseq____"^;logseq____",logseq____"~u652bbc48-6823-44bb-a76a-8cd243da1f09logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn ergibtlogseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Flogseq____",54,536870919]],[logseq____"^15logseq____",[85,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",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____"~u652bbc48-3349-4466-9f1b-077d310eed69logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Flogseq____",103,536870919]],[logseq____"^15logseq____",[86,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Vlogseq____",83,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[86,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[86,logseq____"^;logseq____",logseq____"~u652bbc48-03c6-491d-b5ad-1b4dae52f6a8logseq____",536870919]],[logseq____"^15logseq____",[87,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____",[87,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Flogseq____",82,536870919]],[logseq____"^15logseq____",[87,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^Vlogseq____",82,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[87,logseq____"^;logseq____",logseq____"~u652bbc48-3430-4d22-adc8-4d3dce1ac111logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Flogseq____",69,536870919]],[logseq____"^15logseq____",[88,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Vlogseq____",48,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[88,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[88,logseq____"^;logseq____",logseq____"~u652bbc48-00c0-44e0-9074-39cd9236e581logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(a+b)^3(a^2+b^2)^3(a-b)^3logseq____&=(a^2+b^2)^3\\\\left((a+b)(a-b)\\\\right)^3\\\\notag\\\\\\\\\\nlogseq____&=(a^2+b^2)^3(a^2-b^2)^3\\\\notag\\\\\\\\\\nlogseq____&=\\\\left((a^2+b^2)(a^2-b^2)\\\\right)^3\\\\notag\\\\\\\\\\nlogseq____&=(a^4-b^4)^3\\\\\\\\\\n\\\\frac{a-b}{(a+b)^{-1}}logseq____&=(a-b)(a+b)\\\\notag\\\\\\\\\\nlogseq____&=a^2-b^2\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Flogseq____",75,536870919]],[logseq____"^15logseq____",[89,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Vlogseq____",75,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[89,logseq____"^;logseq____",logseq____"~u652bbc48-2b27-4965-8a17-25a6f55dc165logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Flogseq____",76,536870919]],[logseq____"^15logseq____",[90,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[90,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[90,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[90,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[90,logseq____"^;logseq____",logseq____"~u652bbc48-89e9-4878-ab7a-fd04952b383blogseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Qlogseq____",logseq____"# [[Potenzen]] und [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Flogseq____",64,536870919]],[logseq____"^15logseq____",[91,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[91,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[91,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[91,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[91,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[91,logseq____"^;logseq____",logseq____"~u652bbc48-8182-4961-9ac9-60ded1609649logseq____",536870919]],[logseq____"^15logseq____",[92,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____",[92,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[92,logseq____"^Flogseq____",67,536870919]],[logseq____"^15logseq____",[92,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^Vlogseq____",67,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[92,logseq____"^;logseq____",logseq____"~u652bbc48-7ebd-40d2-9d1b-ecca7a61fd9elogseq____",536870919]],[logseq____"^15logseq____",[93,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____",[93,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Flogseq____",49,536870919]],[logseq____"^15logseq____",[93,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^Vlogseq____",49,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[93,logseq____"^;logseq____",logseq____"~u652bbc48-c996-4dde-8e42-ee6e6dae176alogseq____",536870919]],[logseq____"^15logseq____",[94,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____",[94,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Flogseq____",75,536870919]],[logseq____"^15logseq____",[94,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^Vlogseq____",73,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[94,logseq____"^;logseq____",logseq____"~u652bbc48-36e8-4ca7-b65c-9531ee29616blogseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Qlogseq____",logseq____"#FIXME $(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5=2+20x+40x^2$ ist Falschlogseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Flogseq____",55,536870919]],[logseq____"^15logseq____",[95,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Vlogseq____",88,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[95,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[95,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[95,logseq____"^;logseq____",logseq____"~u652bbc48-20ef-4e11-98f5-e7a405ab5ebclogseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\sqrt{a+5}-\\\\sqrt{x+3}logseq____&=2\\\\sqrt(x+1) logseq____&|logseq____&()^2\\\\\\\\\\n\\\\left(\\\\sqrt{x+5}-\\\\sqrt{x+3}\\\\right)^2logseq____&=4(x+1)\\\\\\\\\\n(x+5)-2\\\\sqrt{x+5}\\\\sqrt{x+3}+(x+3)logseq____&=4x+4logseq____&|logseq____&-(2x+8)\\\\\\\\\\n-2\\\\sqrt{(x+5)(x+3)}logseq____&=2x-4 logseq____&|logseq____&()^2\\\\\\\\\\n4(x+5)(x+3)logseq____&=4x^2-16x+16\\\\\\\\\\n4x^2+32x+60logseq____&=4x^2-16x+16logseq____&|logseq____&-4x^2+16x-16\\\\\\\\\\n48x+44logseq____&=0\\\\quad x=-\\\\frac{11}{12}\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Flogseq____",81,536870919]],[logseq____"^15logseq____",[96,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Vlogseq____",81,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[96,logseq____"^;logseq____",logseq____"~u652bbc48-975c-4397-9387-936f6dde7beelogseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Qlogseq____",logseq____"$\\\\text{Allgemein}$\\n\\\\begin{align}\\n(a+b)^2logseq____&=a^2+2ab+b^2logseq____&,\\\\text{ 1. 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von [Punkten]([[Punkte]]) $(x,y)$ in der [Ebene]([[Ebenen]]) die den gleichen [[Abstand]] r ([[Radius]]) zu einem festen [Punkt]([[Punkte]]) 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1}\\\\quad 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logseq____",logseq____"^18logseq____",2],536870922]],[logseq____"^15logseq____",[191,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[191,logseq____"^Hlogseq____",184,536870922]],[logseq____"^15logseq____",[191,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[191,logseq____"^;logseq____",logseq____"~u652bbc49-9b83-4ea8-a478-0542a2c0ec47logseq____",536870922]],[logseq____"^15logseq____",[192,logseq____"^Qlogseq____",logseq____"Alternativ: $p\\\\equiv q$, falls $p\\\\leftrightarrow q$ [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[192,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[192,logseq____"^Flogseq____",214,536870922]],[logseq____"^15logseq____",[192,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[192,logseq____"^Vlogseq____",191,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[192,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[192,logseq____"^Hlogseq____",174,536870922]],[logseq____"^15logseq____",[192,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[192,logseq____"^;logseq____",logseq____"~u652bbc49-e768-4543-a96b-6839fe3964d4logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Qlogseq____",logseq____"### [[Aussagenlogische Variable]]logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[193,logseq____"^Flogseq____",204,536870922]],[logseq____"^15logseq____",[193,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[193,logseq____"^Vlogseq____",191,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",169,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[193,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[193,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[193,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[193,logseq____"^Hlogseq____",169,536870922]],[logseq____"^15logseq____",[193,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[193,logseq____"^;logseq____",logseq____"~u652bbc49-1b9f-440b-bd50-2fb40eeaf3c0logseq____",536870922]],[logseq____"^15logseq____",[194,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____",[194,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[194,logseq____"^Flogseq____",202,536870922]],[logseq____"^15logseq____",[194,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[194,logseq____"^Vlogseq____",202,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[194,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[194,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[194,logseq____"^;logseq____",logseq____"~u652bbc49-b63a-4047-82b0-5bf36230a7a3logseq____",536870922]],[logseq____"^15logseq____",[195,logseq____"^Qlogseq____",logseq____"[[Wikipedia]]logseq____",536870922]],[logseq____"^15logseq____",[195,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[195,logseq____"^Flogseq____",203,536870922]],[logseq____"^15logseq____",[195,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[195,logseq____"^Vlogseq____",209,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[195,logseq____"^Ulogseq____",170,536870922]],[logseq____"^15logseq____",[195,logseq____"^Hlogseq____",170,536870922]],[logseq____"^15logseq____",[195,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[195,logseq____"^;logseq____",logseq____"~u652bbc49-f8f3-48a1-b75d-e1b30ee4084dlogseq____",536870922]],[logseq____"^15logseq____",[196,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____",[196,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[196,logseq____"^Flogseq____",190,536870922]],[logseq____"^15logseq____",[196,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[196,logseq____"^Vlogseq____",188,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",182,536870922]],[logseq____"^15logseq____",[196,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[196,logseq____"^Hlogseq____",168,536870922]],[logseq____"^15logseq____",[196,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[196,logseq____"^;logseq____",logseq____"~u652bbc49-2ae8-415b-8c9b-c2fbc477f422logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Qlogseq____",logseq____"Formel, die konstant $w$ istlogseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[197,logseq____"^Flogseq____",201,536870922]],[logseq____"^15logseq____",[197,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[197,logseq____"^Vlogseq____",201,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",174,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[197,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[197,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[197,logseq____"^;logseq____",logseq____"~u652bbc49-968f-4731-9ed5-93e81eb801aclogseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Qlogseq____",logseq____"[Variable]([[Variablen]]), die den Wert $w$ oder $f$ annimmtlogseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[198,logseq____"^Flogseq____",193,536870922]],[logseq____"^15logseq____",[198,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[198,logseq____"^Vlogseq____",193,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",169,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[198,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[198,logseq____"^Hlogseq____",176,536870922]],[logseq____"^15logseq____",[198,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[198,logseq____"^;logseq____",logseq____"~u652bbc49-435d-4710-bda3-a74e381a48f0logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Qlogseq____",logseq____"### [[Belegung]] (der [[Variablen]])logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[199,logseq____"^Flogseq____",212,536870922]],[logseq____"^15logseq____",[199,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[199,logseq____"^Vlogseq____",191,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",176,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[199,logseq____"^Ulogseq____",186,536870922]],[logseq____"^15logseq____",[199,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[199,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[199,logseq____"^Hlogseq____",176,536870922]],[logseq____"^15logseq____",[199,logseq____"^Hlogseq____",186,536870922]],[logseq____"^15logseq____",[199,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[199,logseq____"^;logseq____",logseq____"~u652bbc49-d011-4b6f-8750-90f199a6b0cblogseq____",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____",191,536870922]],[logseq____"^15logseq____",[200,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[200,logseq____"^Vlogseq____",191,536870922]],[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____"~u652bbc49-0744-49c7-b04f-d4a4c1cae11flogseq____",536870922]],[logseq____"^15logseq____",[201,logseq____"^Qlogseq____",logseq____"### [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[201,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[201,logseq____"^Flogseq____",188,536870922]],[logseq____"^15logseq____",[201,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[201,logseq____"^Vlogseq____",191,536870922]],[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____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[201,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[201,logseq____"^Hlogseq____",174,536870922]],[logseq____"^15logseq____",[201,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[201,logseq____"^;logseq____",logseq____"~u652bbc49-ab2b-488e-aa5e-b017be5df65dlogseq____",536870922]],[logseq____"^15logseq____",[202,logseq____"^Qlogseq____",logseq____"Weitere [[Beispiele]]logseq____",536870922]],[logseq____"^15logseq____",[202,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[202,logseq____"^Flogseq____",192,536870922]],[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____"^Hlogseq____",35,536870922]],[logseq____"^15logseq____",[202,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[202,logseq____"^;logseq____",logseq____"~u652bbc49-6a7a-471c-8414-14b679bf0456logseq____",536870922]],[logseq____"^15logseq____",[203,logseq____"^Qlogseq____",logseq____"C. Meinel, M. Mundhenk, [[\\logseq____"Mathematische Grundlagen der Informatik\\logseq____", 5. Auflage 2011]]logseq____",536870922]],[logseq____"^15logseq____",[203,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[203,logseq____"^Flogseq____",209,536870922]],[logseq____"^15logseq____",[203,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[203,logseq____"^Vlogseq____",209,536870922]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[203,logseq____"^Ulogseq____",173,536870922]],[logseq____"^15logseq____",[203,logseq____"^Hlogseq____",173,536870922]],[logseq____"^15logseq____",[203,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[203,logseq____"^;logseq____",logseq____"~u652bbc49-bab2-43b2-9c7b-ba27505991aelogseq____",536870922]],[logseq____"^15logseq____",[204,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____",[204,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[204,logseq____"^Flogseq____",213,536870922]],[logseq____"^15logseq____",[204,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[204,logseq____"^Vlogseq____",191,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",172,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",177,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",179,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",183,536870922]],[logseq____"^15logseq____",[204,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[204,logseq____"^Hlogseq____",172,536870922]],[logseq____"^15logseq____",[204,logseq____"^Hlogseq____",177,536870922]],[logseq____"^15logseq____",[204,logseq____"^Hlogseq____",179,536870922]],[logseq____"^15logseq____",[204,logseq____"^Hlogseq____",183,536870922]],[logseq____"^15logseq____",[204,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[204,logseq____"^;logseq____",logseq____"~u652bbc49-5386-491f-a47f-b1c440668c5alogseq____",536870922]],[logseq____"^15logseq____",[205,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____",[205,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[205,logseq____"^Flogseq____",189,536870922]],[logseq____"^15logseq____",[205,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[205,logseq____"^Vlogseq____",187,536870922]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[205,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[205,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[205,logseq____"^;logseq____",logseq____"~u652bbc49-aee0-42fb-ac7e-1b2e63d106eblogseq____",536870922]],[logseq____"^15logseq____",[206,logseq____"^Qlogseq____",logseq____"### [[Kontradiktion]]logseq____",536870922]],[logseq____"^15logseq____",[206,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[206,logseq____"^Flogseq____",201,536870922]],[logseq____"^15logseq____",[206,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[206,logseq____"^Vlogseq____",191,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",175,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[206,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[206,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[206,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[206,logseq____"^Hlogseq____",175,536870922]],[logseq____"^15logseq____",[206,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[206,logseq____"^;logseq____",logseq____"~u652bbc49-8907-4b56-ac1d-e2f5e8552e41logseq____",536870922]],[logseq____"^15logseq____",[207,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____",[207,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[207,logseq____"^Flogseq____",213,536870922]],[logseq____"^15logseq____",[207,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[207,logseq____"^Vlogseq____",213,536870922]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[207,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[207,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[207,logseq____"^;logseq____",logseq____"~u652bbc49-7dac-4c87-887b-cebe38013560logseq____",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____",212,536870922]],[logseq____"^15logseq____",[208,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[208,logseq____"^Vlogseq____",212,536870922]],[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____"~u652bbc49-43e3-4e9b-bb7e-f0b8d5265289logseq____",536870922]],[logseq____"^15logseq____",[209,logseq____"^Qlogseq____",logseq____"# Büchertiplogseq____",536870922]],[logseq____"^15logseq____",[209,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[209,logseq____"^Flogseq____",165,536870922]],[logseq____"^15logseq____",[209,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[209,logseq____"^Vlogseq____",165,536870922]],[logseq____"^15logseq____",[209,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[209,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[209,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[209,logseq____"^17logseq____",false,536870922]],[logseq____"^15logseq____",[209,logseq____"^;logseq____",logseq____"~u652bbc49-5ac0-4e4f-8f23-8aea3d84d954logseq____",536870922]],[logseq____"^15logseq____",[210,logseq____"^Qlogseq____",logseq____"Formel, die konstant $f$ istlogseq____",536870922]],[logseq____"^15logseq____",[210,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[210,logseq____"^Flogseq____",206,536870922]],[logseq____"^15logseq____",[210,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[210,logseq____"^Vlogseq____",206,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",175,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[210,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[210,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[210,logseq____"^;logseq____",logseq____"~u652bbc49-34bb-474c-8d26-603e57bde46dlogseq____",536870922]],[logseq____"^15logseq____",[211,logseq____"^Qlogseq____",logseq____"Zuordnung von $w/f$ an jede [Variable]([[Variablen]]) einer [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[211,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[211,logseq____"^Flogseq____",199,536870922]],[logseq____"^15logseq____",[211,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[211,logseq____"^Vlogseq____",199,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[211,logseq____"^Ulogseq____",176,536870922]],[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____",176,536870922]],[logseq____"^15logseq____",[211,logseq____"^Hlogseq____",182,536870922]],[logseq____"^15logseq____",[211,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[211,logseq____"^;logseq____",logseq____"~u652bbc49-c22d-49cb-a095-ff2bb89d4c6dlogseq____",536870922]],[logseq____"^15logseq____",[212,logseq____"^Qlogseq____",logseq____"### [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[212,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[212,logseq____"^Flogseq____",193,536870922]],[logseq____"^15logseq____",[212,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[212,logseq____"^Vlogseq____",191,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",182,536870922]],[logseq____"^15logseq____",[212,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[212,logseq____"^?logseq____",[logseq____"^ 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$p,q$ [logisch äquivalent]([[Logisch Äquivalent]]), wenn sie den gleichen [[Wahrheitswerteverlauf]] haben.\\n[[Schreibweise]]: $p\\\\equiv q$logseq____",536870922]],[logseq____"^15logseq____",[214,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[214,logseq____"^Flogseq____",206,536870922]],[logseq____"^15logseq____",[214,logseq____"^Xlogseq____",165,536870922]],[logseq____"^15logseq____",[214,logseq____"^Vlogseq____",191,536870922]],[logseq____"^15logseq____",[214,logseq____"^Ulogseq____",165,536870922]],[logseq____"^15logseq____",[214,logseq____"^Ulogseq____",168,536870922]],[logseq____"^15logseq____",[214,logseq____"^Ulogseq____",178,536870922]],[logseq____"^15logseq____",[214,logseq____"^Ulogseq____",180,536870922]],[logseq____"^15logseq____",[214,logseq____"^Ulogseq____",184,536870922]],[logseq____"^15logseq____",[214,logseq____"^Ulogseq____",185,536870922]],[logseq____"^15logseq____",[214,logseq____"^Hlogseq____",168,536870922]],[logseq____"^15logseq____",[214,logseq____"^Hlogseq____",178,536870922]],[logseq____"^15logseq____",[214,logseq____"^Hlogseq____",185,536870922]],[logseq____"^15logseq____",[214,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[214,logseq____"^;logseq____",logseq____"~u652bbc49-719e-46f5-a751-318be89e8497logseq____",536870922]],[logseq____"^15logseq____",[216,logseq____"^Klogseq____",1697365065689,536870923]],[logseq____"^15logseq____",[216,logseq____"^@logseq____",false,536870923]],[logseq____"^15logseq____",[216,logseq____"^Ylogseq____",logseq____"regelnlogseq____",536870923]],[logseq____"^15logseq____",[216,logseq____"^11logseq____",logseq____"Regelnlogseq____",536870923]],[logseq____"^15logseq____",[216,logseq____"^Blogseq____",1697365065689,536870923]],[logseq____"^15logseq____",[216,logseq____"^;logseq____",logseq____"~u652bbc49-97e0-4717-aa78-084d0f1b0161logseq____",536870923]],[logseq____"^15logseq____",[217,logseq____"^Klogseq____",1697365065678,536870923]],[logseq____"^15logseq____",[217,logseq____"^@logseq____",false,536870923]],[logseq____"^15logseq____",[217,logseq____"^Ylogseq____",logseq____"de morgansche regelnlogseq____",536870923]],[logseq____"^15logseq____",[217,logseq____"^11logseq____",logseq____"de Morgansche 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p(x)),\\\\quadlogseq____&\\\\text{Negationsregeln.}\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[245,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[245,logseq____"^Flogseq____",251,536870923]],[logseq____"^15logseq____",[245,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[245,logseq____"^Vlogseq____",251,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",216,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[245,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[245,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[245,logseq____"^;logseq____",logseq____"~u652bbc49-f75e-4bd2-9f00-2a3121d280e3logseq____",536870923]],[logseq____"^15logseq____",[246,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist falsch}\\n\\\\end{align}\\n[Beweis]([[Beweise]]):logseq____",536870923]],[logseq____"^15logseq____",[246,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[246,logseq____"^Flogseq____",267,536870923]],[logseq____"^15logseq____",[246,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[246,logseq____"^Vlogseq____",263,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",227,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[246,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[246,logseq____"^Hlogseq____",227,536870923]],[logseq____"^15logseq____",[246,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[246,logseq____"^;logseq____",logseq____"~u652bbc49-2548-4a0e-a0f5-b16f923f58e6logseq____",536870923]],[logseq____"^15logseq____",[247,logseq____"^Qlogseq____",logseq____"Endliches Universum $U$ mit Objekten $O_1,\\\\cdots,O_m$logseq____",536870923]],[logseq____"^15logseq____",[247,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[247,logseq____"^Flogseq____",248,536870923]],[logseq____"^15logseq____",[247,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[247,logseq____"^Vlogseq____",254,536870923]],[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____"~u652bbc49-2221-43b5-9f6a-9f779e9486d7logseq____",536870923]],[logseq____"^15logseq____",[248,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____",[248,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[248,logseq____"^Flogseq____",243,536870923]],[logseq____"^15logseq____",[248,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[248,logseq____"^Vlogseq____",254,536870923]],[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____"~u652bbc49-403c-46c3-8df8-9b032d14924flogseq____",536870923]],[logseq____"^15logseq____",[249,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____",[249,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[249,logseq____"^Flogseq____",257,536870923]],[logseq____"^15logseq____",[249,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[249,logseq____"^Vlogseq____",257,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[249,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[249,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[249,logseq____"^;logseq____",logseq____"~u652bbc49-e602-45a4-ab2c-cf9f6234fdd8logseq____",536870923]],[logseq____"^15logseq____",[250,logseq____"^Qlogseq____",logseq____"# [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[250,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[250,logseq____"^Flogseq____",223,536870923]],[logseq____"^15logseq____",[250,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[250,logseq____"^Vlogseq____",223,536870923]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",216,536870923]],[logseq____"^15logseq____",[250,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[250,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[250,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[250,logseq____"^Hlogseq____",216,536870923]],[logseq____"^15logseq____",[250,logseq____"^17logseq____",false,536870923]],[logseq____"^15logseq____",[250,logseq____"^;logseq____",logseq____"~u652bbc49-2d4e-4e9d-8206-fac05ea576b5logseq____",536870923]],[logseq____"^15logseq____",[251,logseq____"^Qlogseq____",logseq____"## [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[251,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[251,logseq____"^Flogseq____",265,536870923]],[logseq____"^15logseq____",[251,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[251,logseq____"^Vlogseq____",254,536870923]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",216,536870923]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[251,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[251,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870923]],[logseq____"^15logseq____",[251,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[251,logseq____"^Hlogseq____",216,536870923]],[logseq____"^15logseq____",[251,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[251,logseq____"^;logseq____",logseq____"~u652bbc49-bc9f-4d36-b12d-fe013f972d42logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Qlogseq____",logseq____"Schachtelung von quantoren ([Beispiel]([[Beispiele]])):logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[252,logseq____"^Flogseq____",245,536870923]],[logseq____"^15logseq____",[252,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[252,logseq____"^Vlogseq____",251,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",216,536870923]],[logseq____"^15logseq____",[252,logseq____"^Ulogseq____",223,536870923]],[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____"~u652bbc49-1b5b-4fd0-9157-4d3e27e599balogseq____",536870923]],[logseq____"^15logseq____",[253,logseq____"^Qlogseq____",logseq____"Verschiedene [[Quantoren]] dürfen **nicht** vertauscht werdenlogseq____",536870923]],[logseq____"^15logseq____",[253,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[253,logseq____"^Flogseq____",276,536870923]],[logseq____"^15logseq____",[253,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[253,logseq____"^Vlogseq____",272,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[253,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[253,logseq____"^Hlogseq____",228,536870923]],[logseq____"^15logseq____",[253,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[253,logseq____"^;logseq____",logseq____"~u652bbc49-b26c-4ad3-9007-e1f97a34ab2dlogseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Qlogseq____",logseq____"# [[Aussageform]]logseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[254,logseq____"^Flogseq____",277,536870923]],[logseq____"^15logseq____",[254,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[254,logseq____"^Vlogseq____",223,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[254,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[254,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[254,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[254,logseq____"^Hlogseq____",234,536870923]],[logseq____"^15logseq____",[254,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[254,logseq____"^;logseq____",logseq____"~u652bbc49-34a2-4ef4-8930-256e37b2bad8logseq____",536870923]],[logseq____"^15logseq____",[255,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____",[255,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[255,logseq____"^Flogseq____",247,536870923]],[logseq____"^15logseq____",[255,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[255,logseq____"^Vlogseq____",254,536870923]],[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____"~u652bbc49-8767-4a0d-8390-36f1bc82022flogseq____",536870923]],[logseq____"^15logseq____",[256,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____",[256,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[256,logseq____"^Flogseq____",268,536870923]],[logseq____"^15logseq____",[256,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[256,logseq____"^Vlogseq____",268,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",172,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",180,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[256,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[256,logseq____"^Hlogseq____",172,536870923]],[logseq____"^15logseq____",[256,logseq____"^Hlogseq____",180,536870923]],[logseq____"^15logseq____",[256,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[256,logseq____"^;logseq____",logseq____"~u652bbc49-e6ee-43fe-b72a-2eff1f51ad4dlogseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[257,logseq____"^Flogseq____",264,536870923]],[logseq____"^15logseq____",[257,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[257,logseq____"^Vlogseq____",254,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[257,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[257,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[257,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[257,logseq____"^;logseq____",logseq____"~u652bbc49-e0c1-4dfe-8c3a-ad885da59960logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\logseq____"\\\\text{x ist prim}\\logseq____" \\\\quad logseq____& \\\\text{ist AF mit Variablen x über den natürlichen Zahlen}\\\\\\\\\\n\\logseq____"\\\\underbrace{x}_{x_1}logseq____<\\\\underbrace{y}_{x_2}\\logseq____"\\\\quad logseq____&\\\\text{ist AF mit Variablen }x,y\\\\text{ über die reellen Zahlen } \\\\underbrace{\\\\Reals}_{U_1},\\\\underbrace{\\\\Reals}_{U_2}\\\\\\\\\\n\\logseq____"\\\\text{x ist wahr}\\logseq____"\\\\quadlogseq____&\\\\text{ist AF mit Variablen x über dem Universum der Aussagen}\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[258,logseq____"^Flogseq____",261,536870923]],[logseq____"^15logseq____",[258,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[258,logseq____"^Vlogseq____",254,536870923]],[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____"~u652bbc49-67f3-4305-b884-b2c0a0332f34logseq____",536870923]],[logseq____"^15logseq____",[259,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____",[259,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[259,logseq____"^Flogseq____",258,536870923]],[logseq____"^15logseq____",[259,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[259,logseq____"^Vlogseq____",258,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[259,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[259,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[259,logseq____"^;logseq____",logseq____"~u652bbc49-1076-420e-9217-6a2e73c3c715logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Qlogseq____",logseq____"$\\\\blacksquare$ = Beweisende, auch [q.e.d.]([[Q.E.D.]]), [[w.z.b.w.]], QED, logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[260,logseq____"^Flogseq____",266,536870923]],[logseq____"^15logseq____",[260,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[260,logseq____"^Vlogseq____",266,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",218,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",221,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",227,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[260,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[260,logseq____"^Hlogseq____",218,536870923]],[logseq____"^15logseq____",[260,logseq____"^Hlogseq____",221,536870923]],[logseq____"^15logseq____",[260,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[260,logseq____"^;logseq____",logseq____"~u652bbc49-965d-4e9a-a353-9670791ebbfblogseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[261,logseq____"^Flogseq____",241,536870923]],[logseq____"^15logseq____",[261,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[261,logseq____"^Vlogseq____",254,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[261,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[261,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[261,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[261,logseq____"^;logseq____",logseq____"~u652bbc49-5761-4b96-bcbf-5ed346f7d433logseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Qlogseq____",logseq____"see: [[assoziativ]] [[kommutativ]] [[idenpotent]] [[wechselseitig distributiv]] [[distributiv]] [[de Morgansche Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[262,logseq____"^Flogseq____",277,536870923]],[logseq____"^15logseq____",[262,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[262,logseq____"^Vlogseq____",277,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",217,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",222,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",225,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",226,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",232,536870923]],[logseq____"^15logseq____",[262,logseq____"^Ulogseq____",233,536870923]],[logseq____"^15logseq____",[262,logseq____"^Hlogseq____",217,536870923]],[logseq____"^15logseq____",[262,logseq____"^Hlogseq____",222,536870923]],[logseq____"^15logseq____",[262,logseq____"^Hlogseq____",225,536870923]],[logseq____"^15logseq____",[262,logseq____"^Hlogseq____",226,536870923]],[logseq____"^15logseq____",[262,logseq____"^Hlogseq____",232,536870923]],[logseq____"^15logseq____",[262,logseq____"^Hlogseq____",233,536870923]],[logseq____"^15logseq____",[262,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[262,logseq____"^;logseq____",logseq____"~u652bbc49-7652-4b62-9117-08d68713b0fclogseq____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Qlogseq____",logseq____"Betrachte $\\logseq____"xlogseq____<y\\logseq____"$ als AF über $\\\\Reals, \\\\Reals$logseq____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[263,logseq____"^Flogseq____",273,536870923]],[logseq____"^15logseq____",[263,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[263,logseq____"^Vlogseq____",273,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[263,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[263,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[263,logseq____"^;logseq____",logseq____"~u652bbc49-2a7b-4c63-9d8b-b67438d9e836logseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Qlogseq____",logseq____"[Gleichartige]([[Gleichartig]]) [[Quantoren]] dürfen vertauscht werdenlogseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[264,logseq____"^Flogseq____",251,536870923]],[logseq____"^15logseq____",[264,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[264,logseq____"^Vlogseq____",254,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[264,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[264,logseq____"^Hlogseq____",224,536870923]],[logseq____"^15logseq____",[264,logseq____"^Hlogseq____",228,536870923]],[logseq____"^15logseq____",[264,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[264,logseq____"^;logseq____",logseq____"~u652bbc49-6a94-4e3d-9398-a2e25a659d5elogseq____",536870923]],[logseq____"^15logseq____",[265,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____",[265,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[265,logseq____"^Flogseq____",275,536870923]],[logseq____"^15logseq____",[265,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[265,logseq____"^Vlogseq____",254,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[265,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[265,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[265,logseq____"^;logseq____",logseq____"~u652bbc49-3bf8-4290-a524-b313b7dae822logseq____",536870923]],[logseq____"^15logseq____",[266,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____",[266,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[266,logseq____"^Flogseq____",267,536870923]],[logseq____"^15logseq____",[266,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[266,logseq____"^Vlogseq____",267,536870923]],[logseq____"^15logseq____",[266,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[266,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[266,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[266,logseq____"^Ulogseq____",227,536870923]],[logseq____"^15logseq____",[266,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[266,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[266,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[266,logseq____"^;logseq____",logseq____"~u652bbc49-d45e-4785-ba9d-752b67d747ddlogseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist wahr}\\n\\\\end{align}\\n[Beweis]([[Beweise]]):logseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[267,logseq____"^Flogseq____",263,536870923]],[logseq____"^15logseq____",[267,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[267,logseq____"^Vlogseq____",263,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",227,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[267,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[267,logseq____"^Hlogseq____",227,536870923]],[logseq____"^15logseq____",[267,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[267,logseq____"^;logseq____",logseq____"~u652bbc49-cc6e-4a0d-b52c-b9715d403cf7logseq____",536870923]],[logseq____"^15logseq____",[268,logseq____"^Qlogseq____",logseq____"Sei $p(x)$ AF in der Variablen $x$ über $U$:logseq____",536870923]],[logseq____"^15logseq____",[268,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[268,logseq____"^Flogseq____",258,536870923]],[logseq____"^15logseq____",[268,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[268,logseq____"^Vlogseq____",254,536870923]],[logseq____"^15logseq____",[268,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[268,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[268,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[268,logseq____"^;logseq____",logseq____"~u652bbc49-39e4-47b2-962e-8c6841f79857logseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Qlogseq____",logseq____"Wäre $y$ aus $\\\\Reals$ mit $xlogseq____<y$ für alle $x$ aus $\\\\Reals$, so insbesondere auch $ylogseq____<y$, Widerspruch. $\\\\blacksquare$logseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[269,logseq____"^Flogseq____",246,536870923]],[logseq____"^15logseq____",[269,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[269,logseq____"^Vlogseq____",246,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",224,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",227,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",228,536870923]],[logseq____"^15logseq____",[269,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[269,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[269,logseq____"^;logseq____",logseq____"~u652bbc49-eae7-4cf6-bb60-7fbb1f41532clogseq____",536870923]],[logseq____"^15logseq____",[270,logseq____"^Qlogseq____",logseq____"see: [[Allquantor]] [[Existenzquantor]]logseq____",536870923]],[logseq____"^15logseq____",[270,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[270,logseq____"^Flogseq____",255,536870923]],[logseq____"^15logseq____",[270,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[270,logseq____"^Vlogseq____",255,536870923]],[logseq____"^15logseq____",[270,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[270,logseq____"^Ulogseq____",229,536870923]],[logseq____"^15logseq____",[270,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[270,logseq____"^Ulogseq____",235,536870923]],[logseq____"^15logseq____",[270,logseq____"^Hlogseq____",229,536870923]],[logseq____"^15logseq____",[270,logseq____"^Hlogseq____",235,536870923]],[logseq____"^15logseq____",[270,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[270,logseq____"^;logseq____",logseq____"~u652bbc49-8fff-417f-aeab-af2ebf1ac11alogseq____",536870923]],[logseq____"^15logseq____",[271,logseq____"^Qlogseq____",logseq____"see: [[Große Azzoziativgesetze]] [[Große de Morgansche Regeln]] [[Negationsregeln]]logseq____",536870923]],[logseq____"^15logseq____",[271,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[271,logseq____"^Flogseq____",245,536870923]],[logseq____"^15logseq____",[271,logseq____"^Xlogseq____",223,536870923]],[logseq____"^15logseq____",[271,logseq____"^Vlogseq____",245,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",216,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",219,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",223,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",231,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",234,536870923]],[logseq____"^15logseq____",[271,logseq____"^Ulogseq____",236,536870923]],[logseq____"^15logseq____",[271,logseq____"^Hlogseq____",219,536870923]],[logseq____"^15logseq____",[271,logseq____"^Hlogseq____",231,536870923]],[logseq____"^15logseq____",[271,logseq____"^Hlogseq____",236,536870923]],[logseq____"^15logseq____",[271,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[271,logseq____"^;logseq____",logseq____"~u652bbc49-90d3-4dd9-b41c-377a48cc70f3logseq____",536870923]],[logseq____"^15logseq____",[272,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 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(AF) über [Universen]([[Universen (Logik)]]) 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