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Lösung von } x^n-a=0 \\\\text{ ist } x=\\\\sqrt[n]a$ #FIXME ich glaube hier fehlt logseq____>0 constraintlogseq____",536870919]],[logseq____"^15logseq____",[53,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[53,logseq____"^Flogseq____",75,536870919]],[logseq____"^15logseq____",[53,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[53,logseq____"^Vlogseq____",75,536870919]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[53,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[53,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[53,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[53,logseq____"^;logseq____",logseq____"~u65296e06-9912-421e-b265-89e51222267dlogseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Qlogseq____",logseq____"# [[Binomische Formeln]]logseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Flogseq____",76,536870919]],[logseq____"^15logseq____",[54,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[54,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[54,logseq____"^Hlogseq____",34,536870919]],[logseq____"^15logseq____",[54,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[54,logseq____"^;logseq____",logseq____"~u65296e06-1860-47fa-8dda-3d62d79e4edalogseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Qlogseq____",logseq____"## [[Pascalsches Dreieck]]logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Flogseq____",85,536870919]],[logseq____"^15logseq____",[55,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[55,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[55,logseq____"^Hlogseq____",28,536870919]],[logseq____"^15logseq____",[55,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[55,logseq____"^;logseq____",logseq____"~u65296e06-324e-4163-b034-7350815d2bc8logseq____",536870919]],[logseq____"^15logseq____",[56,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____",[56,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Flogseq____",76,536870919]],[logseq____"^15logseq____",[56,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[56,logseq____"^;logseq____",logseq____"~u65296e06-609b-485d-8b90-67760ef8ed15logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Qlogseq____",logseq____"## [[Binomialkoeffizient]]logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Flogseq____",52,536870919]],[logseq____"^15logseq____",[57,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Vlogseq____",55,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[57,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[57,logseq____"^Hlogseq____",32,536870919]],[logseq____"^15logseq____",[57,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[57,logseq____"^;logseq____",logseq____"~u65296e06-a822-40dd-80eb-e955b682c653logseq____",536870919]],[logseq____"^15logseq____",[58,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____",[58,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Flogseq____",60,536870919]],[logseq____"^15logseq____",[58,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^Vlogseq____",88,536870919]],[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____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[58,logseq____"^;logseq____",logseq____"~u65296e06-024b-41ce-a9bb-73fb14fd5871logseq____",536870919]],[logseq____"^15logseq____",[59,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____",[59,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Flogseq____",89,536870919]],[logseq____"^15logseq____",[59,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Vlogseq____",89,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",33,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Hlogseq____",33,536870919]],[logseq____"^15logseq____",[59,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[59,logseq____"^;logseq____",logseq____"~u65296e06-bb60-4358-b7d0-71829a2d326flogseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Flogseq____",51,536870919]],[logseq____"^15logseq____",[60,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Vlogseq____",88,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[60,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[60,logseq____"^;logseq____",logseq____"~u65296e06-e4de-498f-8ca2-17262c2b9082logseq____",536870919]],[logseq____"^15logseq____",[61,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____",[61,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Flogseq____",72,536870919]],[logseq____"^15logseq____",[61,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^Vlogseq____",104,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[61,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[61,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[61,logseq____"^;logseq____",logseq____"~u65296e06-3141-4c02-99e2-0479eae9f6d5logseq____",536870919]],[logseq____"^15logseq____",[62,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____",[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____",74,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[62,logseq____"^;logseq____",logseq____"~u65296e06-7b91-4e89-992b-9d62801a153alogseq____",536870919]],[logseq____"^15logseq____",[63,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____",[63,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Flogseq____",69,536870919]],[logseq____"^15logseq____",[63,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^Vlogseq____",57,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[63,logseq____"^;logseq____",logseq____"~u65296e06-fe42-46bb-b077-dd5a40af4317logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Flogseq____",72,536870919]],[logseq____"^15logseq____",[64,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^Vlogseq____",72,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[64,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[64,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[64,logseq____"^;logseq____",logseq____"~u65296e06-02b0-4056-b487-922ae095d790logseq____",536870919]],[logseq____"^15logseq____",[65,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____",[65,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Flogseq____",78,536870919]],[logseq____"^15logseq____",[65,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Vlogseq____",78,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[65,logseq____"^;logseq____",logseq____"~u65296e06-f4a7-419e-90cc-b9bc73909005logseq____",536870919]],[logseq____"^15logseq____",[66,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____",[66,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Flogseq____",86,536870919]],[logseq____"^15logseq____",[66,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[66,logseq____"^;logseq____",logseq____"~u65296e06-6c4d-40c2-afec-8d0d7541857flogseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nax^2+bx+clogseq____&=0logseq____&|:a\\\\quadlogseq____&a\\\\ne0\\\\\\\\\\nx^2+\\\\frac ba x + \\\\frac ca logseq____&= 0\\\\\\\\\\nplogseq____&=\\\\frac ba,logseq____&q = \\\\frac ca\\\\\\\\\\nx^2+px+qlogseq____&=0\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Flogseq____",94,536870919]],[logseq____"^15logseq____",[67,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Vlogseq____",94,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[67,logseq____"^;logseq____",logseq____"~u65296e06-5676-40bc-bd09-a2f361fac007logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\sqrt[n]{a^n}logseq____&=1^\\\\frac nn=a^1=a\\\\\\\\\\n\\\\sqrt[n]{\\\\frac ab} logseq____&= \\\\frac{\\\\sqrt[n]a}{\\\\sqrt[n]b}\\\\\\\\\\n\\\\left(\\\\sqrt[n]a\\\\right)^mlogseq____&=a^\\\\frac mn\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Flogseq____",53,536870919]],[logseq____"^15logseq____",[68,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Vlogseq____",75,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[68,logseq____"^;logseq____",logseq____"~u65296e06-692b-47bf-ba45-49438a5f3f54logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Qlogseq____",logseq____"Das [[Pascalsche Dreieck]] lässt sich auch mit den entsprechenden Binomialkoeffizienten darstellenlogseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Flogseq____",57,536870919]],[logseq____"^15logseq____",[69,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^Vlogseq____",57,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",38,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^Hlogseq____",38,536870919]],[logseq____"^15logseq____",[69,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[69,logseq____"^;logseq____",logseq____"~u65296e06-4c4c-4682-a31c-2cefe88c7623logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Qlogseq____",logseq____"$\\\\text{Allgemein}$\\n\\\\begin{align}\\n\\\\ln a + \\\\ln b logseq____&= \\\\ln(a*b) \\\\\\\\\\n\\\\ln(-a) logseq____&= \\\\frac1{\\\\ln a}\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Flogseq____",92,536870919]],[logseq____"^15logseq____",[70,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^Vlogseq____",81,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[70,logseq____"^;logseq____",logseq____"~u65296e06-f07e-4adf-bab9-a98fdefcd075logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Qlogseq____",logseq____"#FIXME $(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5=2+20x+40x^2$ ist Falschlogseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Flogseq____",99,536870919]],[logseq____"^15logseq____",[71,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Vlogseq____",52,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[71,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[71,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[71,logseq____"^;logseq____",logseq____"~u65296e06-8163-4216-9716-97cf7b0fda8clogseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Qlogseq____",logseq____"[[Satz von Vieta]]\\n\\\\begin{align}\\nx^2+px+qlogseq____&=0 logseq____&logseq____& \\\\text{habe }x_{1,2}\\\\text{ als Lösung}\\\\\\\\\\nx_1+x_2logseq____&=-p logseq____&logseq____& x_1*x_2=q\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Flogseq____",48,536870919]],[logseq____"^15logseq____",[72,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Vlogseq____",104,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[72,logseq____"^Hlogseq____",41,536870919]],[logseq____"^15logseq____",[72,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[72,logseq____"^;logseq____",logseq____"~u65296e06-0e17-42aa-8f5e-6b052d7dbd27logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Qlogseq____",logseq____"$\\\\text{Allgemein}$\\n\\\\begin{align}\\n(a+b)^2logseq____&=a^2+2ab+b^2logseq____&,\\\\text{ 1. Binomische Formel}\\\\\\\\\\n(a-b)^2logseq____&=a^2-2ab+b^2logseq____&,\\\\text{ 2. Binomische Formel}\\\\\\\\\\n(a+b)(a-b)logseq____&=a^2-b^2logseq____&,\\\\text{ 3. Binomische Formel}\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Flogseq____",54,536870919]],[logseq____"^15logseq____",[73,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[73,logseq____"^;logseq____",logseq____"~u65296e06-c96a-4bb9-b25e-6325cc16d72blogseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Flogseq____",81,536870919]],[logseq____"^15logseq____",[74,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[74,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[74,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[74,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[74,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[74,logseq____"^;logseq____",logseq____"~u65296e06-112d-4dc1-94e5-b60a8458ec8flogseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Qlogseq____",logseq____"# [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Flogseq____",88,536870919]],[logseq____"^15logseq____",[75,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^Vlogseq____",47,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[75,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[75,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[75,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[75,logseq____"^;logseq____",logseq____"~u65296e06-b649-4948-a99e-ca4172b6e6b6logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Trigonometrie]]logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Flogseq____",78,536870919]],[logseq____"^15logseq____",[76,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[76,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[76,logseq____"^Hlogseq____",36,536870919]],[logseq____"^15logseq____",[76,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[76,logseq____"^;logseq____",logseq____"~u65296e06-e95e-4ecc-915f-4be7bc5e34falogseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Qlogseq____",logseq____"Einsetzen, um zu überprüfen, ob man eine Fallunterscheidung brauchtlogseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Flogseq____",66,536870919]],[logseq____"^15logseq____",[77,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^Vlogseq____",86,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[77,logseq____"^;logseq____",logseq____"~u65296e06-7139-4e2e-aff7-bff904de533alogseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Qlogseq____",logseq____"# Was ist Größer?logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Flogseq____",74,536870919]],[logseq____"^15logseq____",[78,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[78,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[78,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[78,logseq____"^;logseq____",logseq____"~u65296e06-ea1d-4a06-a796-b8458b66f7e7logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Qlogseq____",logseq____"## [[Rationalmachen]] des [[Nenner]]slogseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Flogseq____",49,536870919]],[logseq____"^15logseq____",[79,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Vlogseq____",75,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[79,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[79,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[79,logseq____"^Hlogseq____",39,536870919]],[logseq____"^15logseq____",[79,logseq____"^Hlogseq____",45,536870919]],[logseq____"^15logseq____",[79,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[79,logseq____"^;logseq____",logseq____"~u65296e06-e04a-4380-802a-4f0e6c445140logseq____",536870919]],[logseq____"^15logseq____",[80,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____",[80,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Flogseq____",56,536870919]],[logseq____"^15logseq____",[80,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[80,logseq____"^;logseq____",logseq____"~u65296e06-9308-4c12-9691-70e279f710d4logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Logarithmen]]logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Flogseq____",89,536870919]],[logseq____"^15logseq____",[81,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",31,536870919]],[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____",31,536870919]],[logseq____"^15logseq____",[81,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[81,logseq____"^;logseq____",logseq____"~u65296e06-6f26-47c6-ac72-e573b1d87329logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\sqrt{x^4+6x^2y+9y^2}logseq____&=\\\\sqrt{(x^2+3y)^2}=x^2+3y\\\\\\\\\\n\\\\sqrt {x + \\\\sqrt{x^2-y^2}}\\\\sqrt{x-\\\\sqrt{x^2-y^2}}logseq____&=\\\\sqrt{\\\\left(x + \\\\sqrt{x^2-y^2}\\\\right)\\\\left(x-\\\\sqrt{x^2-y^2}\\\\right)}\\\\notag\\\\\\\\\\nlogseq____&=\\\\sqrt{x^2-(x^2-y^2)}\\\\notag\\\\\\\\\\nlogseq____&=\\\\sqrt{y^2}=\\\\left|y\\\\right|\\\\\\\\\\n\\\\sqrt[5]{a\\\\sqrt[3]a}=\\\\sqrt[5]{\\\\sqrt[3]{a^3}\\\\sqrt[3]a}\\nlogseq____&=\\\\sqrt[5]{\\\\sqrt[3]{a^4}}\\n=\\\\sqrt[5]{a^\\\\frac43}\\n=a^{\\\\frac43\\\\frac15}=a^\\\\frac4{15}\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Flogseq____",90,536870919]],[logseq____"^15logseq____",[82,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^Vlogseq____",90,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[82,logseq____"^;logseq____",logseq____"~u65296e06-a049-4d20-a2bf-517a0109b3celogseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Flogseq____",101,536870919]],[logseq____"^15logseq____",[83,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^Vlogseq____",79,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[83,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[83,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[83,logseq____"^;logseq____",logseq____"~u65296e06-9a5f-4f24-b401-dc9312827eb9logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn ergibtlogseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Flogseq____",82,536870919]],[logseq____"^15logseq____",[84,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^Vlogseq____",90,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[84,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[84,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[84,logseq____"^;logseq____",logseq____"~u65296e06-13df-40d5-aaf0-a04110690fd6logseq____",536870919]],[logseq____"^15logseq____",[85,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____",[85,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Flogseq____",73,536870919]],[logseq____"^15logseq____",[85,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[85,logseq____"^;logseq____",logseq____"~u65296e06-1750-42b5-afcc-cf1618e914fdlogseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Qlogseq____",logseq____"# [[Wurzelgleichung]]logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Flogseq____",94,536870919]],[logseq____"^15logseq____",[86,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Vlogseq____",47,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[86,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[86,logseq____"^Hlogseq____",30,536870919]],[logseq____"^15logseq____",[86,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[86,logseq____"^;logseq____",logseq____"~u65296e06-fe77-4bdf-9f05-ae9af6c713dflogseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Qlogseq____",logseq____"logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Flogseq____",59,536870919]],[logseq____"^15logseq____",[87,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^Vlogseq____",89,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[87,logseq____"^;logseq____",logseq____"~u65296e06-6a02-49c5-af41-ad85c3b905e2logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Qlogseq____",logseq____"# [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Flogseq____",47,536870919]],[logseq____"^15logseq____",[88,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Vlogseq____",47,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[88,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[88,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[88,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[88,logseq____"^;logseq____",logseq____"~u65296e06-ea9f-40ce-9ae3-96aa096b9875logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Flogseq____",91,536870919]],[logseq____"^15logseq____",[89,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[89,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[89,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[89,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[89,logseq____"^;logseq____",logseq____"~u65296e06-2925-4799-a26a-0119d06f0e1dlogseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Flogseq____",68,536870919]],[logseq____"^15logseq____",[90,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Vlogseq____",75,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[90,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[90,logseq____"^;logseq____",logseq____"~u65296e06-380a-4b4d-a391-359a5c1edbbalogseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Qlogseq____",logseq____"# [[Links]]logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Flogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[91,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[91,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[91,logseq____"^Hlogseq____",44,536870919]],[logseq____"^15logseq____",[91,logseq____"^17logseq____",false,536870919]],[logseq____"^15logseq____",[91,logseq____"^;logseq____",logseq____"~u65296e06-e5e9-492a-91db-8b7ca35a6e56logseq____",536870919]],[logseq____"^15logseq____",[92,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____",[92,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[92,logseq____"^Flogseq____",81,536870919]],[logseq____"^15logseq____",[92,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^Vlogseq____",81,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[92,logseq____"^;logseq____",logseq____"~u65296e06-67dc-4854-9138-4f9202203160logseq____",536870919]],[logseq____"^15logseq____",[93,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____",[93,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Flogseq____",55,536870919]],[logseq____"^15logseq____",[93,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^Vlogseq____",55,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[93,logseq____"^;logseq____",logseq____"~u65296e06-0a38-4324-aa4a-87d3a64ff07dlogseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Qlogseq____",logseq____"# [[Quadratische Gleichungen]]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____",47,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[94,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[94,logseq____"^Hlogseq____",29,536870919]],[logseq____"^15logseq____",[94,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[94,logseq____"^;logseq____",logseq____"~u65296e06-ffd6-487d-b022-b99ea16283bdlogseq____",536870919]],[logseq____"^15logseq____",[95,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____",[95,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Flogseq____",60,536870919]],[logseq____"^15logseq____",[95,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Vlogseq____",60,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[95,logseq____"^;logseq____",logseq____"~u65296e06-d751-4a4d-be90-23d72f860a64logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Qlogseq____",logseq____"https://www.wolframalpha.comlogseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Flogseq____",103,536870919]],[logseq____"^15logseq____",[96,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Vlogseq____",103,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[96,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[96,logseq____"^;logseq____",logseq____"~u65296e06-1723-478f-ab62-ec1c6a9992d9logseq____",536870919]],[logseq____"^15logseq____",[97,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____",[97,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Flogseq____",74,536870919]],[logseq____"^15logseq____",[97,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[97,logseq____"^Vlogseq____",74,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[97,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[97,logseq____"^;logseq____",logseq____"~u65296e06-6d40-4269-baf0-8e699868cae4logseq____",536870919]],[logseq____"^15logseq____",[98,logseq____"^Qlogseq____",logseq____"#FIXME $\\\\sin^{-2}x+cos^2y=1,r$ ist 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die konstant $w$ istlogseq____",536870922]],[logseq____"^15logseq____",[160,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[160,logseq____"^Flogseq____",185,536870922]],[logseq____"^15logseq____",[160,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[160,logseq____"^Vlogseq____",185,536870922]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",147,536870922]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[160,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[160,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[160,logseq____"^;logseq____",logseq____"~u65296e07-a633-46a8-9dba-57e140c17996logseq____",536870922]],[logseq____"^15logseq____",[161,logseq____"^Qlogseq____",logseq____"Entsteht durch sukzessive [[Verknüpfungen]] wie oben an Variablen ergibt #FIXMElogseq____",536870922]],[logseq____"^15logseq____",[161,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[161,logseq____"^Flogseq____",184,536870922]],[logseq____"^15logseq____",[161,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[161,logseq____"^Vlogseq____",184,536870922]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",42,536870922]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",154,536870922]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",155,536870922]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[161,logseq____"^Hlogseq____",42,536870922]],[logseq____"^15logseq____",[161,logseq____"^Hlogseq____",154,536870922]],[logseq____"^15logseq____",[161,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[161,logseq____"^;logseq____",logseq____"~u65296e07-f364-4c3d-b224-bbcf0328c31flogseq____",536870922]],[logseq____"^15logseq____",[162,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____",[162,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[162,logseq____"^Flogseq____",173,536870922]],[logseq____"^15logseq____",[162,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[162,logseq____"^Vlogseq____",173,536870922]],[logseq____"^15logseq____",[162,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[162,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[162,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[162,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[162,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[162,logseq____"^;logseq____",logseq____"~u65296e07-b6f5-483a-ad3e-b0ac11957426logseq____",536870922]],[logseq____"^15logseq____",[163,logseq____"^Qlogseq____",logseq____"Jede mögliche [[Belegung]] wird ein [[Wahrheitswert]] der [[Formel]] zugeordnet (nach [[Tabellenregeln]])logseq____",536870922]],[logseq____"^15logseq____",[163,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[163,logseq____"^Flogseq____",168,536870922]],[logseq____"^15logseq____",[163,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[163,logseq____"^Vlogseq____",168,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",141,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",144,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",145,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",152,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",155,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[163,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[163,logseq____"^Hlogseq____",144,536870922]],[logseq____"^15logseq____",[163,logseq____"^Hlogseq____",145,536870922]],[logseq____"^15logseq____",[163,logseq____"^Hlogseq____",152,536870922]],[logseq____"^15logseq____",[163,logseq____"^Hlogseq____",159,536870922]],[logseq____"^15logseq____",[163,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[163,logseq____"^;logseq____",logseq____"~u65296e07-1d64-4984-adf2-9251d7bd1622logseq____",536870922]],[logseq____"^15logseq____",[164,logseq____"^Qlogseq____",logseq____"Alternativ: $p\\\\equiv q$, falls $p\\\\leftrightarrow q$ [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[164,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[164,logseq____"^Flogseq____",165,536870922]],[logseq____"^15logseq____",[164,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[164,logseq____"^Vlogseq____",169,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",147,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[164,logseq____"^Hlogseq____",147,536870922]],[logseq____"^15logseq____",[164,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[164,logseq____"^;logseq____",logseq____"~u65296e07-09c4-480c-983c-fe9893d56a09logseq____",536870922]],[logseq____"^15logseq____",[165,logseq____"^Qlogseq____",logseq____"Formeln $p,q$ [[logisch äquivalent]], wenn sie den gleichen [[Wahrheitswerteverlauf]] haben.\\n[[Schreibweise]]: $p\\\\equiv q$logseq____",536870922]],[logseq____"^15logseq____",[165,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[165,logseq____"^Flogseq____",177,536870922]],[logseq____"^15logseq____",[165,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[165,logseq____"^Vlogseq____",169,536870922]],[logseq____"^15logseq____",[165,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[165,logseq____"^Ulogseq____",141,536870922]],[logseq____"^15logseq____",[165,logseq____"^Ulogseq____",151,536870922]],[logseq____"^15logseq____",[165,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[165,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[165,logseq____"^Ulogseq____",158,536870922]],[logseq____"^15logseq____",[165,logseq____"^Hlogseq____",141,536870922]],[logseq____"^15logseq____",[165,logseq____"^Hlogseq____",151,536870922]],[logseq____"^15logseq____",[165,logseq____"^Hlogseq____",158,536870922]],[logseq____"^15logseq____",[165,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[165,logseq____"^;logseq____",logseq____"~u65296e07-2da7-48c4-bf92-07f3de5ccf27logseq____",536870922]],[logseq____"^15logseq____",[166,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____",[166,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[166,logseq____"^Flogseq____",179,536870922]],[logseq____"^15logseq____",[166,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[166,logseq____"^Vlogseq____",169,536870922]],[logseq____"^15logseq____",[166,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[166,logseq____"^Ulogseq____",150,536870922]],[logseq____"^15logseq____",[166,logseq____"^Ulogseq____",152,536870922]],[logseq____"^15logseq____",[166,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[166,logseq____"^Ulogseq____",156,536870922]],[logseq____"^15logseq____",[166,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[166,logseq____"^Hlogseq____",150,536870922]],[logseq____"^15logseq____",[166,logseq____"^Hlogseq____",152,536870922]],[logseq____"^15logseq____",[166,logseq____"^Hlogseq____",156,536870922]],[logseq____"^15logseq____",[166,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[166,logseq____"^;logseq____",logseq____"~u65296e07-35be-4b7b-b27e-35c4bdcce8eblogseq____",536870922]],[logseq____"^15logseq____",[167,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____",[167,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[167,logseq____"^Flogseq____",178,536870922]],[logseq____"^15logseq____",[167,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[167,logseq____"^Vlogseq____",176,536870922]],[logseq____"^15logseq____",[167,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[167,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[167,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[167,logseq____"^;logseq____",logseq____"~u65296e07-ab96-4711-8b98-e4203d3dfbedlogseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^Qlogseq____",logseq____"### [[Wahrheitswerteverlauf]] einer [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^Flogseq____",183,536870922]],[logseq____"^15logseq____",[168,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[168,logseq____"^Vlogseq____",169,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ulogseq____",141,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ulogseq____",155,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[168,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[168,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[168,logseq____"^Hlogseq____",141,536870922]],[logseq____"^15logseq____",[168,logseq____"^Hlogseq____",155,536870922]],[logseq____"^15logseq____",[168,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[168,logseq____"^;logseq____",logseq____"~u65296e07-56a7-4da1-a666-1d97a3844b24logseq____",536870922]],[logseq____"^15logseq____",[169,logseq____"^Qlogseq____",logseq____"## [[Verknüpfung von Aussagen]]logseq____",536870922]],[logseq____"^15logseq____",[169,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[169,logseq____"^Flogseq____",167,536870922]],[logseq____"^15logseq____",[169,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[169,logseq____"^Vlogseq____",176,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[169,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870922]],[logseq____"^15logseq____",[169,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[169,logseq____"^Hlogseq____",157,536870922]],[logseq____"^15logseq____",[169,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[169,logseq____"^;logseq____",logseq____"~u65296e07-6f58-40d2-9ca5-90b69f92d8f0logseq____",536870922]],[logseq____"^15logseq____",[170,logseq____"^Qlogseq____",logseq____"Formel, die konstant $f$ istlogseq____",536870922]],[logseq____"^15logseq____",[170,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[170,logseq____"^Flogseq____",177,536870922]],[logseq____"^15logseq____",[170,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[170,logseq____"^Vlogseq____",177,536870922]],[logseq____"^15logseq____",[170,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[170,logseq____"^Ulogseq____",148,536870922]],[logseq____"^15logseq____",[170,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[170,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[170,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[170,logseq____"^;logseq____",logseq____"~u65296e07-aef6-4979-82a4-7986cd75fed9logseq____",536870922]],[logseq____"^15logseq____",[171,logseq____"^Qlogseq____",logseq____"# Büchertiplogseq____",536870922]],[logseq____"^15logseq____",[171,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[171,logseq____"^Flogseq____",138,536870922]],[logseq____"^15logseq____",[171,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[171,logseq____"^Vlogseq____",138,536870922]],[logseq____"^15logseq____",[171,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[171,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[171,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[171,logseq____"^17logseq____",false,536870922]],[logseq____"^15logseq____",[171,logseq____"^;logseq____",logseq____"~u65296e07-cf2e-472a-8493-c45b140c9852logseq____",536870922]],[logseq____"^15logseq____",[172,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$|\\n|-|-|-|-|-|-|\\n|f|f||w|w|w|w|\\n|f|w||w|f|f|f|\\n|w|f||f|w|f|f|\\n|w|w||w|w|w|w|logseq____",536870922]],[logseq____"^15logseq____",[172,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[172,logseq____"^Flogseq____",163,536870922]],[logseq____"^15logseq____",[172,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[172,logseq____"^Vlogseq____",168,536870922]],[logseq____"^15logseq____",[172,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[172,logseq____"^Ulogseq____",141,536870922]],[logseq____"^15logseq____",[172,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[172,logseq____"^Ulogseq____",155,536870922]],[logseq____"^15logseq____",[172,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[172,logseq____"^Hlogseq____",141,536870922]],[logseq____"^15logseq____",[172,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[172,logseq____"^;logseq____",logseq____"~u65296e07-61e1-4270-a492-467d6338e011logseq____",536870922]],[logseq____"^15logseq____",[173,logseq____"^Qlogseq____",logseq____"Weitere 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$\\\\blacksquare$logseq____",536870923]],[logseq____"^15logseq____",[217,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[217,logseq____"^Flogseq____",224,536870923]],[logseq____"^15logseq____",[217,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[217,logseq____"^Vlogseq____",224,536870923]],[logseq____"^15logseq____",[217,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[217,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[217,logseq____"^Ulogseq____",197,536870923]],[logseq____"^15logseq____",[217,logseq____"^Ulogseq____",200,536870923]],[logseq____"^15logseq____",[217,logseq____"^Ulogseq____",201,536870923]],[logseq____"^15logseq____",[217,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[217,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[217,logseq____"^;logseq____",logseq____"~u65296e07-4a63-4b2d-80c6-80f6ec062aeflogseq____",536870923]],[logseq____"^15logseq____",[218,logseq____"^Qlogseq____",logseq____"Betrachte $\\logseq____"xlogseq____<y\\logseq____"$ als AF über $\\\\Reals, \\\\Reals$logseq____",536870923]],[logseq____"^15logseq____",[218,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[218,logseq____"^Flogseq____",251,536870923]],[logseq____"^15logseq____",[218,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[218,logseq____"^Vlogseq____",251,536870923]],[logseq____"^15logseq____",[218,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[218,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[218,logseq____"^Ulogseq____",197,536870923]],[logseq____"^15logseq____",[218,logseq____"^Ulogseq____",201,536870923]],[logseq____"^15logseq____",[218,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[218,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[218,logseq____"^;logseq____",logseq____"~u65296e07-9665-4c8b-9d3b-946f1d9b176flogseq____",536870923]],[logseq____"^15logseq____",[219,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(p\\\\land q)\\\\land r logseq____&\\\\equiv p\\\\land(q\\\\land r),\\\\quadlogseq____&\\\\text{schreibe daher: } p\\\\land q\\\\land r,\\\\quad \\\\land \\\\text{ assoziativ}\\\\\\\\\\n(p \\\\lor q) \\\\lor r logseq____&\\\\equiv p \\\\lor (q\\\\lor r),\\\\quadlogseq____&\\\\text{schreibe daher: } p\\\\lor q\\\\lor r,\\\\quad \\\\lor \\\\text{ assoziativ}\\\\\\\\\\np \\\\land q logseq____&\\\\equiv q \\\\land p,\\\\quad \\\\land logseq____&\\\\text{ kommutativ}\\\\\\\\\\np \\\\lor q logseq____&\\\\equiv q \\\\lor p,\\\\quad \\\\lor logseq____&\\\\text{ kommutativ}\\\\\\\\\\n\\\\neg(\\\\neg p) logseq____&\\\\equiv p, \\\\quad \\\\neg logseq____&\\\\text{ idenpotent}\\\\\\\\\\np\\\\land(q\\\\lor r)logseq____&\\\\equiv (p\\\\land q)\\\\lor(p\\\\land r);\\\\notag\\\\\\\\\\np\\\\lor(q\\\\land r)logseq____&\\\\equiv (p\\\\lor q)\\\\lor(p\\\\land r),\\\\quad logseq____&\\\\land,\\\\lor\\\\text{ wechselseitig distributiv}\\\\\\\\\\n\\\\neg(p\\\\land q)logseq____&\\\\equiv(\\\\neg p)\\\\lor(\\\\neg q);\\\\notag\\\\\\\\\\n\\\\neg(p\\\\land q)logseq____&\\\\equiv(\\\\neg p)\\\\land (\\\\neg q), \\\\quad logseq____&\\\\text{ (kleine) de Morgansche Regeln}\\\\\\\\\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[219,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[219,logseq____"^Flogseq____",223,536870923]],[logseq____"^15logseq____",[219,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[219,logseq____"^Vlogseq____",196,536870923]],[logseq____"^15logseq____",[219,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[219,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[219,logseq____"^;logseq____",logseq____"~u65296e07-6c24-4161-9b8c-c56d7d760a4elogseq____",536870923]],[logseq____"^15logseq____",[220,logseq____"^Qlogseq____",logseq____"see: [[Allquantor]] [[Existenzquantor]]logseq____",536870923]],[logseq____"^15logseq____",[220,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[220,logseq____"^Flogseq____",234,536870923]],[logseq____"^15logseq____",[220,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[220,logseq____"^Vlogseq____",234,536870923]],[logseq____"^15logseq____",[220,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[220,logseq____"^Ulogseq____",202,536870923]],[logseq____"^15logseq____",[220,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[220,logseq____"^Ulogseq____",208,536870923]],[logseq____"^15logseq____",[220,logseq____"^Hlogseq____",202,536870923]],[logseq____"^15logseq____",[220,logseq____"^Hlogseq____",208,536870923]],[logseq____"^15logseq____",[220,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[220,logseq____"^;logseq____",logseq____"~u65296e07-03f4-40ec-a87a-b0fa0d61a7d0logseq____",536870923]],[logseq____"^15logseq____",[221,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____",[221,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[221,logseq____"^Flogseq____",244,536870923]],[logseq____"^15logseq____",[221,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[221,logseq____"^Vlogseq____",244,536870923]],[logseq____"^15logseq____",[221,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[221,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[221,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[221,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[221,logseq____"^;logseq____",logseq____"~u65296e07-31b9-4de0-8ab4-080e4cefc9f6logseq____",536870923]],[logseq____"^15logseq____",[222,logseq____"^Qlogseq____",logseq____"Die beiden Quantoren $\\\\forall, \\\\exists$ erweitern die [Sprache]([[Sprachen]]) der [[Aussagenlogik]]logseq____",536870923]],[logseq____"^15logseq____",[222,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[222,logseq____"^Flogseq____",235,536870923]],[logseq____"^15logseq____",[222,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[222,logseq____"^Vlogseq____",229,536870923]],[logseq____"^15logseq____",[222,logseq____"^Ulogseq____",193,536870923]],[logseq____"^15logseq____",[222,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[222,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[222,logseq____"^Ulogseq____",212,536870923]],[logseq____"^15logseq____",[222,logseq____"^Hlogseq____",193,536870923]],[logseq____"^15logseq____",[222,logseq____"^Hlogseq____",212,536870923]],[logseq____"^15logseq____",[222,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[222,logseq____"^;logseq____",logseq____"~u65296e07-80cb-4b12-bff1-863ca365ff71logseq____",536870923]],[logseq____"^15logseq____",[223,logseq____"^Qlogseq____",logseq____"# [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[223,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[223,logseq____"^Flogseq____",196,536870923]],[logseq____"^15logseq____",[223,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[223,logseq____"^Vlogseq____",196,536870923]],[logseq____"^15logseq____",[223,logseq____"^Ulogseq____",189,536870923]],[logseq____"^15logseq____",[223,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[223,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[223,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[223,logseq____"^Hlogseq____",189,536870923]],[logseq____"^15logseq____",[223,logseq____"^17logseq____",false,536870923]],[logseq____"^15logseq____",[223,logseq____"^;logseq____",logseq____"~u65296e07-5c0e-4785-b20b-c68262aa74f9logseq____",536870923]],[logseq____"^15logseq____",[224,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____",[224,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[224,logseq____"^Flogseq____",216,536870923]],[logseq____"^15logseq____",[224,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[224,logseq____"^Vlogseq____",218,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",197,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",200,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",201,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[224,logseq____"^Hlogseq____",200,536870923]],[logseq____"^15logseq____",[224,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[224,logseq____"^;logseq____",logseq____"~u65296e07-4b0a-4289-8980-9c32aacfea09logseq____",536870923]],[logseq____"^15logseq____",[225,logseq____"^Qlogseq____",logseq____"wobei $\\\\pi$ eine \\logseq____"[[Permutation]]\\logseq____" der Zahl $(1,2,\\\\cdots,n)$ istlogseq____",536870923]],[logseq____"^15logseq____",[225,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[225,logseq____"^Flogseq____",236,536870923]],[logseq____"^15logseq____",[225,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[225,logseq____"^Vlogseq____",236,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",197,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",201,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",203,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[225,logseq____"^Hlogseq____",203,536870923]],[logseq____"^15logseq____",[225,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[225,logseq____"^;logseq____",logseq____"~u65296e07-b4da-4b45-a3df-6db49d49541blogseq____",536870923]],[logseq____"^15logseq____",[226,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____",[226,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[226,logseq____"^Flogseq____",238,536870923]],[logseq____"^15logseq____",[226,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[226,logseq____"^Vlogseq____",229,536870923]],[logseq____"^15logseq____",[226,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[226,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[226,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[226,logseq____"^;logseq____",logseq____"~u65296e07-4213-4a2e-b2ad-71fa4e8ae472logseq____",536870923]],[logseq____"^15logseq____",[227,logseq____"^Qlogseq____",logseq____"## [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[227,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[227,logseq____"^Flogseq____",247,536870923]],[logseq____"^15logseq____",[227,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[227,logseq____"^Vlogseq____",229,536870923]],[logseq____"^15logseq____",[227,logseq____"^Ulogseq____",189,536870923]],[logseq____"^15logseq____",[227,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[227,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[227,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870923]],[logseq____"^15logseq____",[227,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[227,logseq____"^Hlogseq____",189,536870923]],[logseq____"^15logseq____",[227,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[227,logseq____"^;logseq____",logseq____"~u65296e07-ef71-4142-bdb0-71263e3e8750logseq____",536870923]],[logseq____"^15logseq____",[228,logseq____"^Qlogseq____",logseq____"#FIXMElogseq____",536870923]],[logseq____"^15logseq____",[228,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[228,logseq____"^Flogseq____",247,536870923]],[logseq____"^15logseq____",[228,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[228,logseq____"^Vlogseq____",247,536870923]],[logseq____"^15logseq____",[228,logseq____"^Ulogseq____",42,536870923]],[logseq____"^15logseq____",[228,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[228,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[228,logseq____"^Hlogseq____",42,536870923]],[logseq____"^15logseq____",[228,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[228,logseq____"^;logseq____",logseq____"~u65296e07-a28c-4592-b00c-175000aa1421logseq____",536870923]],[logseq____"^15logseq____",[229,logseq____"^Qlogseq____",logseq____"# [[Aussageform]]logseq____",536870923]],[logseq____"^15logseq____",[229,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[229,logseq____"^Flogseq____",219,536870923]],[logseq____"^15logseq____",[229,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[229,logseq____"^Vlogseq____",196,536870923]],[logseq____"^15logseq____",[229,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[229,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[229,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[229,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[229,logseq____"^Hlogseq____",207,536870923]],[logseq____"^15logseq____",[229,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[229,logseq____"^;logseq____",logseq____"~u65296e07-0c4f-4404-a500-d677a3b864a4logseq____",536870923]],[logseq____"^15logseq____",[230,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____",[230,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[230,logseq____"^Flogseq____",216,536870923]],[logseq____"^15logseq____",[230,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[230,logseq____"^Vlogseq____",216,536870923]],[logseq____"^15logseq____",[230,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[230,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[230,logseq____"^Ulogseq____",197,536870923]],[logseq____"^15logseq____",[230,logseq____"^Ulogseq____",200,536870923]],[logseq____"^15logseq____",[230,logseq____"^Ulogseq____",201,536870923]],[logseq____"^15logseq____",[230,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[230,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[230,logseq____"^;logseq____",logseq____"~u65296e07-5e6a-4640-b07c-94e5a76c2620logseq____",536870923]],[logseq____"^15logseq____",[231,logseq____"^Qlogseq____",logseq____"[[Aussageform]] (AF) über [Universen]([[Universen (Logik)]]) $U_1,\\\\cdots,U_n$:logseq____",536870923]],[logseq____"^15logseq____",[231,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[231,logseq____"^Flogseq____",229,536870923]],[logseq____"^15logseq____",[231,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[231,logseq____"^Vlogseq____",229,536870923]],[logseq____"^15logseq____",[231,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[231,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[231,logseq____"^Ulogseq____",211,536870923]],[logseq____"^15logseq____",[231,logseq____"^Hlogseq____",207,536870923]],[logseq____"^15logseq____",[231,logseq____"^Hlogseq____",211,536870923]],[logseq____"^15logseq____",[231,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[231,logseq____"^;logseq____",logseq____"~u65296e07-b8fc-44eb-94ca-f3f244d42b16logseq____",536870923]],[logseq____"^15logseq____",[232,logseq____"^Qlogseq____",logseq____"$\\\\exists x:p(x)$ ist die Aussage: \\logseq____"Es gibt (wenigstens) ein x aus U für das $p(x)$ wahr ist\\nIhr [[Wahrheitswert]] ist $\\\\begin{cases}w \\\\quadlogseq____&\\\\text{falls }p(x)\\\\text{ w ist für (wenigstens) ein x aus U}\\\\\\\\f\\\\quad logseq____&\\\\text{falls }p(x)\\\\text{ f ist für alle x aus 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\\\\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____",[234,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[234,logseq____"^Flogseq____",215,536870923]],[logseq____"^15logseq____",[234,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[234,logseq____"^Vlogseq____",229,536870923]],[logseq____"^15logseq____",[234,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[234,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[234,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[234,logseq____"^;logseq____",logseq____"~u65296e07-05ae-49ff-b256-4794827ac1eelogseq____",536870923]],[logseq____"^15logseq____",[235,logseq____"^Qlogseq____",logseq____"Sei $p(x)$ AF in der 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x:p(x,y)\\\\\\\\\\nlogseq____&[\\\\forall x_1:\\\\forall x_2: \\\\cdots:\\\\forall 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[[Quantoren]] dürfen vertauscht 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das leere Universum gilt\\n\\\\begin{align}\\n\\\\forall x: p(x)\\\\quadlogseq____&Tautologie\\\\\\\\\\n\\\\exists x: p(x)\\\\quadlogseq____&Kontradiktion\\\\\\\\\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[240,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[240,logseq____"^Flogseq____",222,536870923]],[logseq____"^15logseq____",[240,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[240,logseq____"^Vlogseq____",229,536870923]],[logseq____"^15logseq____",[240,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[240,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[240,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[240,logseq____"^;logseq____",logseq____"~u65296e07-e36b-4b73-9e7e-0745d7af8736logseq____",536870923]],[logseq____"^15logseq____",[241,logseq____"^Qlogseq____",logseq____"$\\\\blacksquare$ = Beweisende, auch [q.e.d.]([[Q.E.D.]]), [[w.z.b.w.]], QED, 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[[assoziativ]] [[kommutativ]] [[idenpotent]] [[wechselseitig distributiv]] [[distributiv]] [[de Morgansche Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[242,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[242,logseq____"^Flogseq____",219,536870923]],[logseq____"^15logseq____",[242,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[242,logseq____"^Vlogseq____",219,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",190,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",198,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",199,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",205,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",206,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",190,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",195,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",198,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",199,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",205,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",206,536870923]],[logseq____"^15logseq____",[242,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[242,logseq____"^;logseq____",logseq____"~u65296e07-221d-4aca-ac3c-33054744a8f2logseq____",536870923]],[logseq____"^15logseq____",[243,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____",[243,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[243,logseq____"^Flogseq____",226,536870923]],[logseq____"^15logseq____",[243,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[243,logseq____"^Vlogseq____",226,536870923]],[logseq____"^15logseq____",[243,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[243,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[243,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[243,logseq____"^;logseq____",logseq____"~u65296e07-d085-49d2-9c41-d633386de90elogseq____",536870923]],[logseq____"^15logseq____",[244,logseq____"^Qlogseq____",logseq____"[Beispiel]([[Beispiele]])logseq____",536870923]],[logseq____"^15logseq____",[244,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[244,logseq____"^Flogseq____",237,536870923]],[logseq____"^15logseq____",[244,logseq____"^Xlogseq____",196,536870923]],[logseq____"^15logseq____",[244,logseq____"^Vlogseq____",229,536870923]],[logseq____"^15logseq____",[244,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[244,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[244,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[244,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[244,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[244,logseq____"^;logseq____",logseq____"~u65296e07-07aa-49ec-b0da-76e21bdb4813logseq____",536870923]],[logseq____"^15logseq____",[245,logseq____"^Qlogseq____",logseq____"Verschiedene [[Quantoren]] dürfen **nicht** vertauscht 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