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62 lines
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[[Wurzelgleichung]]logseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[54,logseq____"^Flogseq____",52,536870919]],[logseq____"^15logseq____",[54,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[54,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[54,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[54,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[54,logseq____"^Hlogseq____",30,536870919]],[logseq____"^15logseq____",[54,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[54,logseq____"^;logseq____",logseq____"~u652944af-3585-4edc-9b17-0b5a88dada33logseq____",536870919]],[logseq____"^15logseq____",[55,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____",[55,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[55,logseq____"^Flogseq____",65,536870919]],[logseq____"^15logseq____",[55,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^Vlogseq____",97,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[55,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[55,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[55,logseq____"^;logseq____",logseq____"~u652944af-b837-4bc7-aeda-d35fc3a1100elogseq____",536870919]],[logseq____"^15logseq____",[56,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____",[56,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[56,logseq____"^Flogseq____",74,536870919]],[logseq____"^15logseq____",[56,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^Vlogseq____",74,536870919]],[logseq____"^15logseq____",[56,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[56,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[56,logseq____"^;logseq____",logseq____"~u652944af-22f4-40c4-85ff-9a54825d87b1logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Qlogseq____",logseq____"$\\\\text{Wenn }alogseq____>0\\\\text{, dann}$\\n\\\\begin{align}\\n\\\\frac b{\\\\sqrt a} = \\\\frac b{\\\\sqrt a}*\\\\frac{\\\\sqrt a}{\\\\sqrt a} = \\\\frac {b\\\\sqrt a}a\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[57,logseq____"^Flogseq____",70,536870919]],[logseq____"^15logseq____",[57,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Vlogseq____",70,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[57,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[57,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[57,logseq____"^;logseq____",logseq____"~u652944af-84ef-4d45-bf02-46942898c26clogseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Qlogseq____",logseq____"## [[Binomialkoeffizient]]logseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[58,logseq____"^Flogseq____",59,536870919]],[logseq____"^15logseq____",[58,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^Vlogseq____",50,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[58,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[58,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[58,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[58,logseq____"^Hlogseq____",32,536870919]],[logseq____"^15logseq____",[58,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[58,logseq____"^;logseq____",logseq____"~u652944af-fe9d-4ec2-9fd0-5be23d2cd708logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[59,logseq____"^Flogseq____",48,536870919]],[logseq____"^15logseq____",[59,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Vlogseq____",50,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[59,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[59,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[59,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[59,logseq____"^;logseq____",logseq____"~u652944af-4925-4f81-9787-67ee8f99499blogseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Qlogseq____",logseq____"[[Wolframalpha]]logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[60,logseq____"^Flogseq____",98,536870919]],[logseq____"^15logseq____",[60,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Vlogseq____",98,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[60,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[60,logseq____"^Hlogseq____",46,536870919]],[logseq____"^15logseq____",[60,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[60,logseq____"^;logseq____",logseq____"~u652944af-bfbc-45b0-ba2b-ec4be5629bbalogseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Qlogseq____",logseq____"cosa -logseq____> beliebige Variable\\ncubus -logseq____> hoch 3logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[61,logseq____"^Flogseq____",81,536870919]],[logseq____"^15logseq____",[61,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^Vlogseq____",81,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[61,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[61,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[61,logseq____"^;logseq____",logseq____"~u652944af-01d1-4c3e-9a1f-8545efc43232logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Qlogseq____",logseq____"# Wissensabgleich [[Trigonometrie]]logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[62,logseq____"^Flogseq____",74,536870919]],[logseq____"^15logseq____",[62,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[62,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[62,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[62,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[62,logseq____"^Hlogseq____",36,536870919]],[logseq____"^15logseq____",[62,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[62,logseq____"^;logseq____",logseq____"~u652944af-3aa2-4864-bfef-edfc097c4c9alogseq____",536870919]],[logseq____"^15logseq____",[63,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____",[63,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[63,logseq____"^Flogseq____",52,536870919]],[logseq____"^15logseq____",[63,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^Vlogseq____",52,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[63,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[63,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[63,logseq____"^;logseq____",logseq____"~u652944af-5a4b-439b-adf9-ae58c9d5b84clogseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n(2x+y)^4logseq____&=(2x)^4+4*(2x)^3y+6*(2x)^2y^2+4*2xy^3+y^4\\\\notag\\\\\\\\\\nlogseq____&=16x^4+32x^3y+24x^2y^2+8xy^3+y^4\\\\\\\\\\n(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5logseq____&=2+20x+40x^2\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[64,logseq____"^Flogseq____",59,536870919]],[logseq____"^15logseq____",[64,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^Vlogseq____",59,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[64,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[64,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[64,logseq____"^;logseq____",logseq____"~u652944af-ed5a-498a-a2f8-748d9a8d3656logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Qlogseq____",logseq____"[[Satz von Vieta]]\\n\\\\begin{align}\\nx^2+px+qlogseq____&=0 logseq____&logseq____& \\\\text{habe }x_{1,2}\\\\text{ als Lösung}\\\\\\\\\\nx_1+x_2logseq____&=-p logseq____&logseq____& x_1*x_2=q\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[65,logseq____"^Flogseq____",88,536870919]],[logseq____"^15logseq____",[65,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Vlogseq____",97,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[65,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[65,logseq____"^Hlogseq____",41,536870919]],[logseq____"^15logseq____",[65,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[65,logseq____"^;logseq____",logseq____"~u652944af-64e7-4cde-9a5b-ba94c803c654logseq____",536870919]],[logseq____"^15logseq____",[66,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____",[66,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[66,logseq____"^Flogseq____",85,536870919]],[logseq____"^15logseq____",[66,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^Vlogseq____",62,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[66,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[66,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[66,logseq____"^;logseq____",logseq____"~u652944af-3178-4582-82d2-dd6a4dddff48logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Qlogseq____",logseq____"# [[Binomische Formeln]]logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[67,logseq____"^Flogseq____",62,536870919]],[logseq____"^15logseq____",[67,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[67,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[67,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[67,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[67,logseq____"^Hlogseq____",34,536870919]],[logseq____"^15logseq____",[67,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[67,logseq____"^;logseq____",logseq____"~u652944af-36e9-4238-ad71-aee3c8012062logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[68,logseq____"^Flogseq____",57,536870919]],[logseq____"^15logseq____",[68,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Vlogseq____",70,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[68,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[68,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[68,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[68,logseq____"^;logseq____",logseq____"~u652944af-eb85-4bfd-b1fc-8c72700520f3logseq____",536870919]],[logseq____"^15logseq____",[69,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____",[69,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[69,logseq____"^Flogseq____",104,536870919]],[logseq____"^15logseq____",[69,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^Vlogseq____",104,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[69,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[69,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[69,logseq____"^;logseq____",logseq____"~u652944af-901e-4aad-8f29-d09c6b2f894blogseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Qlogseq____",logseq____"## [[Rationalmachen]] des [[Nenner]]slogseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[70,logseq____"^Flogseq____",73,536870919]],[logseq____"^15logseq____",[70,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^Vlogseq____",93,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[70,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[70,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[70,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[70,logseq____"^Hlogseq____",39,536870919]],[logseq____"^15logseq____",[70,logseq____"^Hlogseq____",45,536870919]],[logseq____"^15logseq____",[70,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[70,logseq____"^;logseq____",logseq____"~u652944af-4bfa-44df-a49a-f8fc0932aaedlogseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Qlogseq____",logseq____"#FIXME $\\\\sin^{-2}x+cos^2y=1,r$ ist falschlogseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[71,logseq____"^Flogseq____",66,536870919]],[logseq____"^15logseq____",[71,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Vlogseq____",66,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[71,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[71,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[71,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[71,logseq____"^;logseq____",logseq____"~u652944af-388b-40cc-ad25-8b4c916bd37elogseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Qlogseq____",logseq____"![draws/2023-10-08-21-23-31.excalidraw](../assets/excalidraw_svg/2023-10-08-21-23-31.svg)logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[72,logseq____"^Flogseq____",96,536870919]],[logseq____"^15logseq____",[72,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^Vlogseq____",51,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[72,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[72,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[72,logseq____"^;logseq____",logseq____"~u652944af-5ce6-4330-af8e-4bb5af39a694logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Qlogseq____",logseq____"$\\\\text{Allgemein}$\\n\\\\begin{align}\\n|y|logseq____&=\\\\begin{cases}\\ny logseq____&\\\\text{ falls } y\\\\ge0\\\\\\\\\\n-y logseq____&\\\\text{ falls } ylogseq____<0\\\\\\\\\\n\\\\end{cases}\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[73,logseq____"^Flogseq____",104,536870919]],[logseq____"^15logseq____",[73,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^Vlogseq____",93,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[73,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[73,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[73,logseq____"^;logseq____",logseq____"~u652944af-a02a-4231-98d1-67c5bc8471d9logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Qlogseq____",logseq____"# Was ist Größer?logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[74,logseq____"^Flogseq____",101,536870919]],[logseq____"^15logseq____",[74,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[74,logseq____"^Vlogseq____",40,536870919]],[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____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[74,logseq____"^;logseq____",logseq____"~u652944af-19d3-44be-8ced-fa84a874dcfblogseq____",536870919]],[logseq____"^15logseq____",[75,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____",[75,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[75,logseq____"^Flogseq____",78,536870919]],[logseq____"^15logseq____",[75,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^Vlogseq____",47,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[75,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[75,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[75,logseq____"^;logseq____",logseq____"~u652944af-1ffa-4703-a471-4c022fc28089logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Qlogseq____",logseq____"# [[Potenzen]] und [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[76,logseq____"^Flogseq____",67,536870919]],[logseq____"^15logseq____",[76,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[76,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[76,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[76,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[76,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[76,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[76,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[76,logseq____"^;logseq____",logseq____"~u652944af-8982-4b1e-9557-aa0ebd5ed716logseq____",536870919]],[logseq____"^15logseq____",[77,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____",[77,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[77,logseq____"^Flogseq____",90,536870919]],[logseq____"^15logseq____",[77,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^Vlogseq____",93,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[77,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[77,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[77,logseq____"^;logseq____",logseq____"~u652944af-a323-4191-818c-c3a5315ac236logseq____",536870919]],[logseq____"^15logseq____",[78,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____",[78,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[78,logseq____"^Flogseq____",47,536870919]],[logseq____"^15logseq____",[78,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^Vlogseq____",47,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",31,536870919]],[logseq____"^15logseq____",[78,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[78,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[78,logseq____"^;logseq____",logseq____"~u652944af-84a8-4127-a27a-aa42499ce442logseq____",536870919]],[logseq____"^15logseq____",[79,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____",[79,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[79,logseq____"^Flogseq____",54,536870919]],[logseq____"^15logseq____",[79,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",30,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[79,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[79,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[79,logseq____"^;logseq____",logseq____"~u652944af-d44e-4667-9b2b-0f072d2dd418logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Qlogseq____",logseq____"# [[Potenzen]]logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[80,logseq____"^Flogseq____",76,536870919]],[logseq____"^15logseq____",[80,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[80,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[80,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[80,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[80,logseq____"^Hlogseq____",27,536870919]],[logseq____"^15logseq____",[80,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[80,logseq____"^;logseq____",logseq____"~u652944af-c74e-4236-9384-ab5c86735bb4logseq____",536870919]],[logseq____"^15logseq____",[81,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____",[81,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[81,logseq____"^Flogseq____",91,536870919]],[logseq____"^15logseq____",[81,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^Vlogseq____",80,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[81,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[81,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[81,logseq____"^;logseq____",logseq____"~u652944af-a5ae-4454-b23b-ff9bcbb6a41elogseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Qlogseq____",logseq____"Einsetzen, um zu überprüfen, ob man eine Fallunterscheidung brauchtlogseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[82,logseq____"^Flogseq____",79,536870919]],[logseq____"^15logseq____",[82,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[82,logseq____"^Vlogseq____",54,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[82,logseq____"^Ulogseq____",30,536870919]],[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____"~u652944af-b775-4b79-baf9-91cf60371ba2logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Qlogseq____",logseq____"[Beispiel](Beispiele)logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[83,logseq____"^Flogseq____",65,536870919]],[logseq____"^15logseq____",[83,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^Vlogseq____",65,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[83,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[83,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[83,logseq____"^;logseq____",logseq____"~u652944af-9617-4c3a-9c76-1fecd09bc5felogseq____",536870919]],[logseq____"^15logseq____",[84,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____",[84,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[84,logseq____"^Flogseq____",67,536870919]],[logseq____"^15logseq____",[84,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^Vlogseq____",67,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[84,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[84,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[84,logseq____"^;logseq____",logseq____"~u652944af-81c8-43c9-950d-1c7139dc36d7logseq____",536870919]],[logseq____"^15logseq____",[85,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____",[85,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[85,logseq____"^Flogseq____",62,536870919]],[logseq____"^15logseq____",[85,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^Vlogseq____",62,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",36,536870919]],[logseq____"^15logseq____",[85,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[85,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[85,logseq____"^;logseq____",logseq____"~u652944af-4b6b-495c-883f-cbba01a727falogseq____",536870919]],[logseq____"^15logseq____",[86,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____",[86,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[86,logseq____"^Flogseq____",84,536870919]],[logseq____"^15logseq____",[86,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^Vlogseq____",67,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[86,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[86,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[86,logseq____"^;logseq____",logseq____"~u652944af-3933-4499-b968-92e4591785cdlogseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Qlogseq____",logseq____"https://www.wolframalpha.comlogseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[87,logseq____"^Flogseq____",60,536870919]],[logseq____"^15logseq____",[87,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^Vlogseq____",60,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[87,logseq____"^Ulogseq____",46,536870919]],[logseq____"^15logseq____",[87,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[87,logseq____"^;logseq____",logseq____"~u652944af-3843-4e94-bc34-384b8a95eb2blogseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Qlogseq____",logseq____"$\\\\text{Herleitung mit }$[[Quadratische Ergänzung]]\\n\\\\begin{align}\\nx^2+px+qlogseq____&=0logseq____&|logseq____&-q\\\\\\\\\\nx^2+pxlogseq____&=-qlogseq____&|logseq____&+\\\\left(\\\\frac p2\\\\right)^2\\\\\\\\\\nx^2+px+\\\\left(\\\\frac p2\\\\right)^2logseq____&=-q+\\\\left(\\\\frac p2\\\\right)^2\\\\\\\\\\n\\\\underbrace{\\\\left(x+\\\\frac p2\\\\right)^2}_\\\\text{1. binomische Formel}logseq____&=-q+\\\\left(\\\\frac p2\\\\right)^2logseq____&|logseq____&\\\\sqrt{()}\\\\\\\\\\n\\\\left|x+\\\\frac p2\\\\right|logseq____&=\\\\sqrt{\\\\frac{p^2}4-q}logseq____&,logseq____&\\\\text{ falls }\\\\frac{p^2}4-q\\\\ge0\\\\\\\\\\nx+\\\\frac p2logseq____&=\\\\pm\\\\sqrt{\\\\frac{p^2}4-q}logseq____&|logseq____&-\\\\frac p2\\\\\\\\\\nxlogseq____&=-\\\\frac p2\\\\pm\\\\sqrt{\\\\frac{p^2}4-q}\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[88,logseq____"^Flogseq____",97,536870919]],[logseq____"^15logseq____",[88,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Vlogseq____",97,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",26,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[88,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[88,logseq____"^Hlogseq____",26,536870919]],[logseq____"^15logseq____",[88,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[88,logseq____"^;logseq____",logseq____"~u652944af-5df3-437a-b716-cd54d9e8f968logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Qlogseq____",logseq____"$\\\\text{Welche quadratische Gleichung hat die Lösung}$\\n$x_1=3\\\\quad x_2=4$\\n\\\\begin{align}\\nx_1+x_2 = 7 logseq____&= p logseq____&, x_1x_2=12=q\\\\\\\\\\nx^2+px+qlogseq____&=0\\\\\\\\\\nx^2-7x+12logseq____&=0\\\\\\\\\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[89,logseq____"^Flogseq____",83,536870919]],[logseq____"^15logseq____",[89,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Vlogseq____",83,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",41,536870919]],[logseq____"^15logseq____",[89,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[89,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[89,logseq____"^;logseq____",logseq____"~u652944af-8c9a-46ed-af92-5ee521300ccclogseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Qlogseq____",logseq____"$\\\\text{Die Lösung von } x^n-a=0 \\\\text{ ist } x=\\\\sqrt[n]a$ #FIXME ich glaube hier fehlt logseq____>0 constraintlogseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[90,logseq____"^Flogseq____",93,536870919]],[logseq____"^15logseq____",[90,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Vlogseq____",93,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[90,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[90,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[90,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[90,logseq____"^;logseq____",logseq____"~u652944af-0329-4ec9-9cd2-179654a003falogseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[91,logseq____"^Flogseq____",102,536870919]],[logseq____"^15logseq____",[91,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Vlogseq____",80,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[91,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[91,logseq____"^Hlogseq____",35,536870919]],[logseq____"^15logseq____",[91,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[91,logseq____"^;logseq____",logseq____"~u652944af-a74b-4f32-87c4-93e7c99378d2logseq____",536870919]],[logseq____"^15logseq____",[92,logseq____"^Qlogseq____",logseq____"Das [[Pascalsche Dreieck]] lässt sich auch mit den entsprechenden Binomialkoeffizienten darstellenlogseq____",536870919]],[logseq____"^15logseq____",[92,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[92,logseq____"^Flogseq____",58,536870919]],[logseq____"^15logseq____",[92,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^Vlogseq____",58,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",32,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",38,536870919]],[logseq____"^15logseq____",[92,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[92,logseq____"^Hlogseq____",38,536870919]],[logseq____"^15logseq____",[92,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[92,logseq____"^;logseq____",logseq____"~u652944af-cc11-4db2-a824-4d2601d0e69blogseq____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Qlogseq____",logseq____"# [[Wurzeln]]logseq____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[93,logseq____"^Flogseq____",80,536870919]],[logseq____"^15logseq____",[93,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^Vlogseq____",76,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[93,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[93,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[93,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[93,logseq____"^Hlogseq____",37,536870919]],[logseq____"^15logseq____",[93,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[93,logseq____"^;logseq____",logseq____"~u652944af-2fb6-413e-bb2d-b518cf6ec315logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\underbrace{\\\\frac6{2+\\\\sqrt2}*\\\\frac{2-\\\\sqrt2}{2-\\\\sqrt2}\\n= \\\\frac{6(2-\\\\sqrt2)}{2^2-(\\\\sqrt2)^2}}_\\\\text{3. binomische Formel}\\nlogseq____&=\\\\frac{6(2-\\\\sqrt2)}2=3(2-\\\\sqrt2)\\\\\\\\\\n\\\\frac{a-b}{\\\\sqrt a+\\\\sqrt b}logseq____&=\\\\frac{a-b}{\\\\sqrt a + \\\\sqrt b}\\\\frac{\\\\sqrt a-\\\\sqrt b}{\\\\sqrt a-\\\\sqrt b}logseq____&, a,b,logseq____>0\\\\notag\\\\\\\\\\n=\\\\frac{(a-b)(\\\\sqrt a-\\\\sqrt b)}{(a-b)}logseq____&=\\\\sqrt a\\\\sqrt b\\n\\\\end{align}logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[94,logseq____"^Flogseq____",68,536870919]],[logseq____"^15logseq____",[94,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^Vlogseq____",68,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",39,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[94,logseq____"^Ulogseq____",45,536870919]],[logseq____"^15logseq____",[94,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[94,logseq____"^;logseq____",logseq____"~u652944af-4f5f-4d36-b4e3-6992e91796falogseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Qlogseq____",logseq____"#FIXME $(1+\\\\sqrt x)^5*(1-\\\\sqrt x)^5=2+20x+40x^2$ ist Falschlogseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[95,logseq____"^Flogseq____",64,536870919]],[logseq____"^15logseq____",[95,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Vlogseq____",59,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",28,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",34,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",35,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[95,logseq____"^Ulogseq____",42,536870919]],[logseq____"^15logseq____",[95,logseq____"^Hlogseq____",42,536870919]],[logseq____"^15logseq____",[95,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[95,logseq____"^;logseq____",logseq____"~u652944af-da2f-4175-bb04-4669ac47b467logseq____",536870919]],[logseq____"^15logseq____",[96,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____",[96,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[96,logseq____"^Flogseq____",51,536870919]],[logseq____"^15logseq____",[96,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Vlogseq____",51,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",33,536870919]],[logseq____"^15logseq____",[96,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[96,logseq____"^Hlogseq____",33,536870919]],[logseq____"^15logseq____",[96,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[96,logseq____"^;logseq____",logseq____"~u652944af-7eb0-47ee-bac4-1f200e06bd9alogseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Qlogseq____",logseq____"## [[pq-Formel]]logseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[97,logseq____"^Flogseq____",63,536870919]],[logseq____"^15logseq____",[97,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[97,logseq____"^Vlogseq____",52,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",27,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",29,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",37,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[97,logseq____"^Ulogseq____",43,536870919]],[logseq____"^15logseq____",[97,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",2],536870919]],[logseq____"^15logseq____",[97,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[97,logseq____"^Hlogseq____",43,536870919]],[logseq____"^15logseq____",[97,logseq____"^17logseq____",true,536870919]],[logseq____"^15logseq____",[97,logseq____"^;logseq____",logseq____"~u652944af-12c2-4d69-8169-6a8e39f2dbfflogseq____",536870919]],[logseq____"^15logseq____",[98,logseq____"^Qlogseq____",logseq____"# [[Links]]logseq____",536870919]],[logseq____"^15logseq____",[98,logseq____"^Ologseq____",logseq____"^16logseq____",536870919]],[logseq____"^15logseq____",[98,logseq____"^Flogseq____",40,536870919]],[logseq____"^15logseq____",[98,logseq____"^Xlogseq____",40,536870919]],[logseq____"^15logseq____",[98,logseq____"^Vlogseq____",40,536870919]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",40,536870919]],[logseq____"^15logseq____",[98,logseq____"^Ulogseq____",44,536870919]],[logseq____"^15logseq____",[98,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870919]],[logseq____"^15logseq____",[98,logseq____"^Jlogseq____",[],536870919]],[logseq____"^15logseq____",[98,logseq____"^Hlogseq____",44,536870919]],[logseq____"^15logseq____",[98,logseq____"^17logseq____",false,536870919]],[logseq____"^15logseq____",[98,logseq____"^;logseq____",logseq____"~u652944af-83f2-4261-82f0-e723988ed8balogseq____",536870919]],[logseq____"^15logseq____",[99,logseq____"^Qlogseq____",logseq____"#FIXME gleichung (50) hat angeblich noch ne Zeile $=x^0+3ylogseq____>0$ welche keinen Sinn 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[[Aussagen]]logseq____",536870922]],[logseq____"^15logseq____",[161,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[161,logseq____"^Flogseq____",183,536870922]],[logseq____"^15logseq____",[161,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[161,logseq____"^Vlogseq____",138,536870922]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[161,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[161,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[161,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[161,logseq____"^Hlogseq____",153,536870922]],[logseq____"^15logseq____",[161,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[161,logseq____"^;logseq____",logseq____"~u652944b0-5702-401f-ba08-9f3aca8e5c43logseq____",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 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logseq____",logseq____"^18logseq____",2],536870922]],[logseq____"^15logseq____",[163,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[163,logseq____"^Hlogseq____",157,536870922]],[logseq____"^15logseq____",[163,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[163,logseq____"^;logseq____",logseq____"~u652944b0-770b-4050-987e-f9430761cdc4logseq____",536870922]],[logseq____"^15logseq____",[164,logseq____"^Qlogseq____",logseq____"Formeln $p,q$ [[logisch äquivalent]], wenn sie den gleichen [[Wahrheitswerteverlauf]] haben.\\n[[Schreibweise]]: $p\\\\equiv q$logseq____",536870922]],[logseq____"^15logseq____",[164,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[164,logseq____"^Flogseq____",171,536870922]],[logseq____"^15logseq____",[164,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[164,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",141,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",151,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[164,logseq____"^Ulogseq____",158,536870922]],[logseq____"^15logseq____",[164,logseq____"^Hlogseq____",141,536870922]],[logseq____"^15logseq____",[164,logseq____"^Hlogseq____",151,536870922]],[logseq____"^15logseq____",[164,logseq____"^Hlogseq____",158,536870922]],[logseq____"^15logseq____",[164,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[164,logseq____"^;logseq____",logseq____"~u652944b0-de9d-48ca-820d-1f50981434e1logseq____",536870922]],[logseq____"^15logseq____",[165,logseq____"^Qlogseq____",logseq____"[[Wikipedia]]logseq____",536870922]],[logseq____"^15logseq____",[165,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[165,logseq____"^Flogseq____",179,536870922]],[logseq____"^15logseq____",[165,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[165,logseq____"^Vlogseq____",183,536870922]],[logseq____"^15logseq____",[165,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[165,logseq____"^Ulogseq____",143,536870922]],[logseq____"^15logseq____",[165,logseq____"^Hlogseq____",143,536870922]],[logseq____"^15logseq____",[165,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[165,logseq____"^;logseq____",logseq____"~u652944b0-1da7-4c34-9e53-9368f87ee11clogseq____",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____",181,536870922]],[logseq____"^15logseq____",[166,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[166,logseq____"^Vlogseq____",163,536870922]],[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____"~u652944b0-e1aa-4d29-82c8-9cfbe86e8c63logseq____",536870922]],[logseq____"^15logseq____",[167,logseq____"^Qlogseq____",logseq____"### [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[167,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[167,logseq____"^Flogseq____",175,536870922]],[logseq____"^15logseq____",[167,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[167,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[167,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[167,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[167,logseq____"^Ulogseq____",155,536870922]],[logseq____"^15logseq____",[167,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[167,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[167,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[167,logseq____"^Hlogseq____",155,536870922]],[logseq____"^15logseq____",[167,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[167,logseq____"^;logseq____",logseq____"~u652944b0-ef74-4fa4-a6fd-0fd6785449ealogseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^Qlogseq____",logseq____"Satz, der wahr oder falsch ist, d.h. der [[Wahrheitswert]] $w$ bzw $f$ hat ($t/f$, $1/0$, $\\\\text{wahr}/\\\\text{falsch}$)logseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[168,logseq____"^Flogseq____",161,536870922]],[logseq____"^15logseq____",[168,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[168,logseq____"^Vlogseq____",161,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ulogseq____",145,536870922]],[logseq____"^15logseq____",[168,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[168,logseq____"^Hlogseq____",145,536870922]],[logseq____"^15logseq____",[168,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[168,logseq____"^;logseq____",logseq____"~u652944b0-3c46-40fb-8cd1-0bd6187d6d35logseq____",536870922]],[logseq____"^15logseq____",[169,logseq____"^Qlogseq____",logseq____"### [[Belegung]] (der [[Variablen]])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____",163,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ulogseq____",149,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[169,logseq____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[169,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[169,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[169,logseq____"^Hlogseq____",149,536870922]],[logseq____"^15logseq____",[169,logseq____"^Hlogseq____",159,536870922]],[logseq____"^15logseq____",[169,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[169,logseq____"^;logseq____",logseq____"~u652944b0-c9ba-4b4d-8d1d-749b40808f69logseq____",536870922]],[logseq____"^15logseq____",[170,logseq____"^Qlogseq____",logseq____"Entsteht durch sukzessive [[Verknüpfungen]] wie oben an Variablen ergibt 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[[Kontradiktion]]logseq____",536870922]],[logseq____"^15logseq____",[171,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[171,logseq____"^Flogseq____",174,536870922]],[logseq____"^15logseq____",[171,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[171,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[171,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[171,logseq____"^Ulogseq____",148,536870922]],[logseq____"^15logseq____",[171,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[171,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[171,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[171,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[171,logseq____"^Hlogseq____",148,536870922]],[logseq____"^15logseq____",[171,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[171,logseq____"^;logseq____",logseq____"~u652944b0-85ba-4d41-8fde-b98fd507c5d9logseq____",536870922]],[logseq____"^15logseq____",[172,logseq____"^Qlogseq____",logseq____"Zuordnung von $w/f$ an jede [Variable](Variablen) einer [[Aussageformel]]logseq____",536870922]],[logseq____"^15logseq____",[172,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[172,logseq____"^Flogseq____",169,536870922]],[logseq____"^15logseq____",[172,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[172,logseq____"^Vlogseq____",169,536870922]],[logseq____"^15logseq____",[172,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[172,logseq____"^Ulogseq____",149,536870922]],[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____"^Ulogseq____",159,536870922]],[logseq____"^15logseq____",[172,logseq____"^Hlogseq____",155,536870922]],[logseq____"^15logseq____",[172,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[172,logseq____"^;logseq____",logseq____"~u652944b0-fc2c-4321-a463-4565b990bc42logseq____",536870922]],[logseq____"^15logseq____",[173,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\np\\\\land q\\\\lor r\\\\\\\\\\n(p\\\\land q)\\\\lor r\\\\\\\\\\np\\\\land (q\\\\lor 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[[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[174,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[174,logseq____"^Flogseq____",160,536870922]],[logseq____"^15logseq____",[174,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[174,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[174,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[174,logseq____"^Ulogseq____",147,536870922]],[logseq____"^15logseq____",[174,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[174,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[174,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[174,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[174,logseq____"^Hlogseq____",147,536870922]],[logseq____"^15logseq____",[174,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[174,logseq____"^;logseq____",logseq____"~u652944b0-89a9-4f95-aefd-16f058f3fbbelogseq____",536870922]],[logseq____"^15logseq____",[175,logseq____"^Qlogseq____",logseq____"### [[Aussagenlogische Variable]]logseq____",536870922]],[logseq____"^15logseq____",[175,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[175,logseq____"^Flogseq____",166,536870922]],[logseq____"^15logseq____",[175,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[175,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[175,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[175,logseq____"^Ulogseq____",142,536870922]],[logseq____"^15logseq____",[175,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[175,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[175,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",3],536870922]],[logseq____"^15logseq____",[175,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[175,logseq____"^Hlogseq____",142,536870922]],[logseq____"^15logseq____",[175,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[175,logseq____"^;logseq____",logseq____"~u652944b0-7e7c-49d2-8782-9148cca13342logseq____",536870922]],[logseq____"^15logseq____",[176,logseq____"^Qlogseq____",logseq____"Sind $p$ und $q$ [[Aussagen]], so auch\\n\\\\begin{align}\\n(p\\\\land q) logseq____&\\\\quadlogseq____& (\\logseq____"\\\\text{und}\\logseq____")\\\\\\\\\\n(p\\\\lor q) logseq____&logseq____& (\\logseq____"\\\\text{oder}\\logseq____")\\\\\\\\\\n(\\\\neg p) logseq____&logseq____& (\\logseq____"\\\\text{nicht}\\logseq____")\\\\\\\\\\n(p\\\\rightarrow q) logseq____&logseq____& (\\logseq____"\\\\text{impliziert}\\logseq____")\\\\\\\\\\n(p\\\\leftrightarrow q) logseq____&logseq____& (\\logseq____"\\\\text{genau dann wenn}\\logseq____")\\n\\\\end{align}logseq____",536870922]],[logseq____"^15logseq____",[176,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[176,logseq____"^Flogseq____",163,536870922]],[logseq____"^15logseq____",[176,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[176,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[176,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[176,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[176,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[176,logseq____"^Hlogseq____",153,536870922]],[logseq____"^15logseq____",[176,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[176,logseq____"^;logseq____",logseq____"~u652944b0-14db-4c5e-9058-3ecbc954fad0logseq____",536870922]],[logseq____"^15logseq____",[177,logseq____"^Qlogseq____",logseq____"Alternativ: $p\\\\equiv q$, falls $p\\\\leftrightarrow q$ [[Tantologie]]logseq____",536870922]],[logseq____"^15logseq____",[177,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[177,logseq____"^Flogseq____",164,536870922]],[logseq____"^15logseq____",[177,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[177,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[177,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[177,logseq____"^Ulogseq____",147,536870922]],[logseq____"^15logseq____",[177,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[177,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[177,logseq____"^Hlogseq____",147,536870922]],[logseq____"^15logseq____",[177,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[177,logseq____"^;logseq____",logseq____"~u652944b0-c907-42b3-9e44-7f2173501c73logseq____",536870922]],[logseq____"^15logseq____",[178,logseq____"^Qlogseq____",logseq____"[Variable](Variablen), die den Wert $w$ oder $f$ annimmtlogseq____",536870922]],[logseq____"^15logseq____",[178,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[178,logseq____"^Flogseq____",175,536870922]],[logseq____"^15logseq____",[178,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[178,logseq____"^Vlogseq____",175,536870922]],[logseq____"^15logseq____",[178,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[178,logseq____"^Ulogseq____",142,536870922]],[logseq____"^15logseq____",[178,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[178,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[178,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[178,logseq____"^;logseq____",logseq____"~u652944b0-2c47-4274-814d-e6505bb99224logseq____",536870922]],[logseq____"^15logseq____",[179,logseq____"^Qlogseq____",logseq____"C. Meinel, M. Mundhenk, [[\\logseq____"Mathematische Grundlagen der Informatik\\logseq____", 5. Auflage 2011]]logseq____",536870922]],[logseq____"^15logseq____",[179,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[179,logseq____"^Flogseq____",183,536870922]],[logseq____"^15logseq____",[179,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[179,logseq____"^Vlogseq____",183,536870922]],[logseq____"^15logseq____",[179,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[179,logseq____"^Ulogseq____",146,536870922]],[logseq____"^15logseq____",[179,logseq____"^Hlogseq____",146,536870922]],[logseq____"^15logseq____",[179,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[179,logseq____"^;logseq____",logseq____"~u652944b0-7d57-4afe-9ee7-ec0369754324logseq____",536870922]],[logseq____"^15logseq____",[180,logseq____"^Qlogseq____",logseq____"Formel, die konstant $w$ istlogseq____",536870922]],[logseq____"^15logseq____",[180,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[180,logseq____"^Flogseq____",174,536870922]],[logseq____"^15logseq____",[180,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[180,logseq____"^Vlogseq____",174,536870922]],[logseq____"^15logseq____",[180,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[180,logseq____"^Ulogseq____",147,536870922]],[logseq____"^15logseq____",[180,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[180,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[180,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[180,logseq____"^;logseq____",logseq____"~u652944b0-d326-48f2-bcf0-ebd482b856f1logseq____",536870922]],[logseq____"^15logseq____",[181,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870922]],[logseq____"^15logseq____",[181,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[181,logseq____"^Flogseq____",176,536870922]],[logseq____"^15logseq____",[181,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[181,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[181,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[181,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[181,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[181,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[181,logseq____"^Hlogseq____",35,536870922]],[logseq____"^15logseq____",[181,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[181,logseq____"^;logseq____",logseq____"~u652944b0-d307-4cad-85b2-740208bd63e3logseq____",536870922]],[logseq____"^15logseq____",[182,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____",[182,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[182,logseq____"^Flogseq____",187,536870922]],[logseq____"^15logseq____",[182,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[182,logseq____"^Vlogseq____",160,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",141,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",155,536870922]],[logseq____"^15logseq____",[182,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[182,logseq____"^Hlogseq____",141,536870922]],[logseq____"^15logseq____",[182,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[182,logseq____"^;logseq____",logseq____"~u652944b0-1818-44a7-a920-076bd60a3436logseq____",536870922]],[logseq____"^15logseq____",[183,logseq____"^Qlogseq____",logseq____"# Büchertiplogseq____",536870922]],[logseq____"^15logseq____",[183,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[183,logseq____"^Flogseq____",138,536870922]],[logseq____"^15logseq____",[183,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[183,logseq____"^Vlogseq____",138,536870922]],[logseq____"^15logseq____",[183,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[183,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870922]],[logseq____"^15logseq____",[183,logseq____"^Jlogseq____",[],536870922]],[logseq____"^15logseq____",[183,logseq____"^17logseq____",false,536870922]],[logseq____"^15logseq____",[183,logseq____"^;logseq____",logseq____"~u652944b0-cc3c-4ffc-8e10-54dfc76bd6cblogseq____",536870922]],[logseq____"^15logseq____",[184,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____",[184,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[184,logseq____"^Flogseq____",168,536870922]],[logseq____"^15logseq____",[184,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[184,logseq____"^Vlogseq____",161,536870922]],[logseq____"^15logseq____",[184,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[184,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[184,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[184,logseq____"^;logseq____",logseq____"~u652944b0-cf23-44c2-b12d-374585326b2blogseq____",536870922]],[logseq____"^15logseq____",[185,logseq____"^Qlogseq____",logseq____"Weitere [[Beispiele]]logseq____",536870922]],[logseq____"^15logseq____",[185,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[185,logseq____"^Flogseq____",177,536870922]],[logseq____"^15logseq____",[185,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[185,logseq____"^Vlogseq____",163,536870922]],[logseq____"^15logseq____",[185,logseq____"^Ulogseq____",35,536870922]],[logseq____"^15logseq____",[185,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[185,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[185,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[185,logseq____"^Hlogseq____",35,536870922]],[logseq____"^15logseq____",[185,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[185,logseq____"^;logseq____",logseq____"~u652944b0-ead3-409d-b962-29de357a86eelogseq____",536870922]],[logseq____"^15logseq____",[186,logseq____"^Qlogseq____",logseq____"Formel, die konstant $f$ istlogseq____",536870922]],[logseq____"^15logseq____",[186,logseq____"^Ologseq____",logseq____"^16logseq____",536870922]],[logseq____"^15logseq____",[186,logseq____"^Flogseq____",171,536870922]],[logseq____"^15logseq____",[186,logseq____"^Xlogseq____",138,536870922]],[logseq____"^15logseq____",[186,logseq____"^Vlogseq____",171,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",138,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",148,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",153,536870922]],[logseq____"^15logseq____",[186,logseq____"^Ulogseq____",157,536870922]],[logseq____"^15logseq____",[186,logseq____"^17logseq____",true,536870922]],[logseq____"^15logseq____",[186,logseq____"^;logseq____",logseq____"~u652944b0-0ec2-4030-9c8e-72430b27b0celogseq____",536870922]],[logseq____"^15logseq____",[187,logseq____"^Qlogseq____",logseq____"Jede mögliche [[Belegung]] wird ein [[Wahrheitswert]] der [[Formel]] zugeordnet (nach 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q);\\\\notag\\\\\\\\\\n\\\\neg(p\\\\land q)logseq____&\\\\equiv(\\\\neg p)\\\\land (\\\\neg q), \\\\quad logseq____&\\\\text{ (kleine) de Morgansche Regeln}\\\\\\\\\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[217,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[217,logseq____"^Flogseq____",218,536870923]],[logseq____"^15logseq____",[217,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[217,logseq____"^Vlogseq____",195,536870923]],[logseq____"^15logseq____",[217,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[217,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[217,logseq____"^;logseq____",logseq____"~u652944b0-4ab3-4422-a80e-058a22ef64bblogseq____",536870923]],[logseq____"^15logseq____",[218,logseq____"^Qlogseq____",logseq____"# [[Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[218,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[218,logseq____"^Flogseq____",195,536870923]],[logseq____"^15logseq____",[218,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[218,logseq____"^Vlogseq____",195,536870923]],[logseq____"^15logseq____",[218,logseq____"^Ulogseq____",189,536870923]],[logseq____"^15logseq____",[218,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[218,logseq____"^?logseq____",[logseq____"^ logseq____",logseq____"^18logseq____",1],536870923]],[logseq____"^15logseq____",[218,logseq____"^Jlogseq____",[],536870923]],[logseq____"^15logseq____",[218,logseq____"^Hlogseq____",189,536870923]],[logseq____"^15logseq____",[218,logseq____"^17logseq____",false,536870923]],[logseq____"^15logseq____",[218,logseq____"^;logseq____",logseq____"~u652944b0-86da-4a6b-84be-912bf99534c0logseq____",536870923]],[logseq____"^15logseq____",[219,logseq____"^Qlogseq____",logseq____"Betrachte $\\logseq____"xlogseq____<y\\logseq____"$ als AF über $\\\\Reals, \\\\Reals$logseq____",536870923]],[logseq____"^15logseq____",[219,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[219,logseq____"^Flogseq____",215,536870923]],[logseq____"^15logseq____",[219,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[219,logseq____"^Vlogseq____",215,536870923]],[logseq____"^15logseq____",[219,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[219,logseq____"^Ulogseq____",198,536870923]],[logseq____"^15logseq____",[219,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[219,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[219,logseq____"^;logseq____",logseq____"~u652944b0-0b41-4b1c-a5f2-d19bff6c017clogseq____",536870923]],[logseq____"^15logseq____",[220,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____",[220,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[220,logseq____"^Flogseq____",233,536870923]],[logseq____"^15logseq____",[220,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[220,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[220,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[220,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[220,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[220,logseq____"^;logseq____",logseq____"~u652944b0-b965-40f3-9df0-1441250145d5logseq____",536870923]],[logseq____"^15logseq____",[221,logseq____"^Qlogseq____",logseq____"Verschiedene [[Quantoren]] dürfen **nicht** vertauscht werdenlogseq____",536870923]],[logseq____"^15logseq____",[221,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[221,logseq____"^Flogseq____",222,536870923]],[logseq____"^15logseq____",[221,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[221,logseq____"^Vlogseq____",232,536870923]],[logseq____"^15logseq____",[221,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[221,logseq____"^Ulogseq____",198,536870923]],[logseq____"^15logseq____",[221,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[221,logseq____"^Hlogseq____",198,536870923]],[logseq____"^15logseq____",[221,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[221,logseq____"^;logseq____",logseq____"~u652944b0-239b-4a32-87d1-882c5741ee1dlogseq____",536870923]],[logseq____"^15logseq____",[222,logseq____"^Qlogseq____",logseq____"wobei $\\\\pi$ eine \\logseq____"[[Permutation]]\\logseq____" der Zahl $(1,2,\\\\cdots,n)$ 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$y$ aus $\\\\Reals$ mit $xlogseq____<y$ für alle $x$ aus $\\\\Reals$, so insbesondere auch $ylogseq____<y$, Widerspruch. $\\\\blacksquare$logseq____",536870923]],[logseq____"^15logseq____",[224,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[224,logseq____"^Flogseq____",244,536870923]],[logseq____"^15logseq____",[224,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[224,logseq____"^Vlogseq____",244,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",198,536870923]],[logseq____"^15logseq____",[224,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[224,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[224,logseq____"^;logseq____",logseq____"~u652944b0-265a-434a-9fa7-7e45621f4dc3logseq____",536870923]],[logseq____"^15logseq____",[225,logseq____"^Qlogseq____",logseq____"$\\\\blacksquare$ = Beweisende, auch [q.e.d.](Q.E.D.), [[w.z.b.w.]], QED, ![image.png](../assets/image_1697192491775_0.png)logseq____",536870923]],[logseq____"^15logseq____",[225,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[225,logseq____"^Flogseq____",208,536870923]],[logseq____"^15logseq____",[225,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[225,logseq____"^Vlogseq____",208,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",193,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",198,536870923]],[logseq____"^15logseq____",[225,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[225,logseq____"^Hlogseq____",193,536870923]],[logseq____"^15logseq____",[225,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[225,logseq____"^;logseq____",logseq____"~u652944b0-cb8b-488b-b668-3d88e0d1a402logseq____",536870923]],[logseq____"^15logseq____",[226,logseq____"^Qlogseq____",logseq____"Endliches Universum $U$ mit Objekten $O_1,\\\\cdots,O_m$logseq____",536870923]],[logseq____"^15logseq____",[226,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[226,logseq____"^Flogseq____",236,536870923]],[logseq____"^15logseq____",[226,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[226,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[226,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[226,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[226,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[226,logseq____"^;logseq____",logseq____"~u652944b0-a072-4420-a515-5b030dfbb57alogseq____",536870923]],[logseq____"^15logseq____",[227,logseq____"^Qlogseq____",logseq____"[Beispiel](Beispiele)logseq____",536870923]],[logseq____"^15logseq____",[227,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[227,logseq____"^Flogseq____",211,536870923]],[logseq____"^15logseq____",[227,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[227,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[227,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[227,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[227,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[227,logseq____"^;logseq____",logseq____"~u652944b0-e2eb-4c8a-aa87-b782fd133512logseq____",536870923]],[logseq____"^15logseq____",[228,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:(p(x)\\\\land q(x)) logseq____&\\\\equiv (\\\\forall x: p(x)) \\\\land (\\\\forall x:q(x));\\\\notag\\\\\\\\\\n\\\\exists x: (p(x) \\\\lor q(x)) logseq____&\\\\equiv (\\\\exists x: p(x))\\\\lor(\\\\exists x: q(x)),\\\\quadlogseq____&\\\\text{Große Assoziativgesetze}\\\\\\\\\\n\\\\neg(\\\\forall x:p(x))logseq____&\\\\equiv \\\\exists x: (\\\\neg p(x));\\\\quadlogseq____&\\\\text{Große de Morgansche Regeln;}\\\\notag\\\\\\\\\\n\\\\neg(\\\\exists x: p(x)) logseq____&\\\\equiv \\\\forall x: (\\\\neg p(x)),\\\\quadlogseq____&\\\\text{Negationsregeln.}\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[228,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[228,logseq____"^Flogseq____",209,536870923]],[logseq____"^15logseq____",[228,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[228,logseq____"^Vlogseq____",209,536870923]],[logseq____"^15logseq____",[228,logseq____"^Ulogseq____",189,536870923]],[logseq____"^15logseq____",[228,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[228,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[228,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[228,logseq____"^;logseq____",logseq____"~u652944b0-f45d-4395-a469-9db5b6f4348dlogseq____",536870923]],[logseq____"^15logseq____",[229,logseq____"^Qlogseq____",logseq____"Die beiden Quantoren $\\\\forall, \\\\exists$ erweitern die [Sprache](Sprachen) der [[Aussagenlogik]]logseq____",536870923]],[logseq____"^15logseq____",[229,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[229,logseq____"^Flogseq____",212,536870923]],[logseq____"^15logseq____",[229,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[229,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[229,logseq____"^Ulogseq____",192,536870923]],[logseq____"^15logseq____",[229,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[229,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[229,logseq____"^Hlogseq____",192,536870923]],[logseq____"^15logseq____",[229,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[229,logseq____"^;logseq____",logseq____"~u652944b0-dffc-4ddc-816f-29bdb40176f8logseq____",536870923]],[logseq____"^15logseq____",[230,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____",[230,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[230,logseq____"^Flogseq____",220,536870923]],[logseq____"^15logseq____",[230,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[230,logseq____"^Vlogseq____",220,536870923]],[logseq____"^15logseq____",[230,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[230,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[230,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[230,logseq____"^;logseq____",logseq____"~u652944b0-1e03-4689-b9b8-89808a07ac0dlogseq____",536870923]],[logseq____"^15logseq____",[231,logseq____"^Qlogseq____",logseq____"see: [[Allquantor]] [[Existenzquantor]]logseq____",536870923]],[logseq____"^15logseq____",[231,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[231,logseq____"^Flogseq____",235,536870923]],[logseq____"^15logseq____",[231,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[231,logseq____"^Vlogseq____",235,536870923]],[logseq____"^15logseq____",[231,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[231,logseq____"^Ulogseq____",199,536870923]],[logseq____"^15logseq____",[231,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[231,logseq____"^Ulogseq____",205,536870923]],[logseq____"^15logseq____",[231,logseq____"^Hlogseq____",199,536870923]],[logseq____"^15logseq____",[231,logseq____"^Hlogseq____",205,536870923]],[logseq____"^15logseq____",[231,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[231,logseq____"^;logseq____",logseq____"~u652944b0-d14a-48fe-b51f-8d59b5c68c28logseq____",536870923]],[logseq____"^15logseq____",[232,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\nlogseq____&\\\\forall x: \\\\forall y: p(x, y) \\\\equiv \\\\forall y: \\\\forall x: p(x,y)\\\\\\\\\\nlogseq____&\\\\exists x:\\\\exists y:p(x,y)\\\\equiv\\\\exists y:\\\\exists x:p(x,y)\\\\\\\\\\nlogseq____&[\\\\forall x_1:\\\\forall x_2: \\\\cdots:\\\\forall x_n:p(x_1,x_2,\\\\cdots,x_n)\\\\notag\\\\\\\\\\nlogseq____&\\\\equiv\\\\forall_{x_{\\\\pi(1)}}:\\\\forall_{x_{\\\\pi(1)}}:\\\\cdots:\\\\forall_{x_{\\\\pi(1)}}:p(x_1,x_2,\\\\cdots,x_n)\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[232,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[232,logseq____"^Flogseq____",211,536870923]],[logseq____"^15logseq____",[232,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[232,logseq____"^Vlogseq____",211,536870923]],[logseq____"^15logseq____",[232,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[232,logseq____"^Ulogseq____",198,536870923]],[logseq____"^15logseq____",[232,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[232,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[232,logseq____"^;logseq____",logseq____"~u652944b0-bf46-4d95-8f2b-e1b1584176dclogseq____",536870923]],[logseq____"^15logseq____",[233,logseq____"^Qlogseq____",logseq____"[[Beispiele]]logseq____",536870923]],[logseq____"^15logseq____",[233,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[233,logseq____"^Flogseq____",237,536870923]],[logseq____"^15logseq____",[233,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[233,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[233,logseq____"^Ulogseq____",35,536870923]],[logseq____"^15logseq____",[233,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[233,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[233,logseq____"^Hlogseq____",35,536870923]],[logseq____"^15logseq____",[233,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[233,logseq____"^;logseq____",logseq____"~u652944b0-87c3-4827-ae31-4ca89001c6adlogseq____",536870923]],[logseq____"^15logseq____",[234,logseq____"^Qlogseq____",logseq____"Sei $p(x,y)$ AF in Variablen $x,y$ über $U_1,U_2$\\n\\\\begin{align}\\n\\\\underbrace{\\\\forall x: \\\\underbrace{\\\\exists y: p(x,y)}_{\\\\text{AF in den Variablen }x\\\\text{ über }U_1}}_{\\\\text{Aussage}}\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[234,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[234,logseq____"^Flogseq____",241,536870923]],[logseq____"^15logseq____",[234,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[234,logseq____"^Vlogseq____",241,536870923]],[logseq____"^15logseq____",[234,logseq____"^Ulogseq____",189,536870923]],[logseq____"^15logseq____",[234,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[234,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[234,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[234,logseq____"^;logseq____",logseq____"~u652944b0-64e0-40a1-b442-6154c3e6da23logseq____",536870923]],[logseq____"^15logseq____",[235,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\underbrace{\\\\forall}_\\\\text{Allquantor} x: p(x) logseq____&\\\\equiv p(O_1) \\\\land p(O_2) \\\\land \\\\cdots \\\\land p(O_m)\\\\\\\\\\n\\\\underbrace{\\\\exists}_\\\\text{Existenzquantor} x: p(x) logseq____&\\\\equiv p(0_1) \\\\lor p(O_2) \\\\lor \\\\cdots \\\\lor p(0_m)\\\\\\\\\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[235,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[235,logseq____"^Flogseq____",226,536870923]],[logseq____"^15logseq____",[235,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[235,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[235,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[235,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[235,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[235,logseq____"^;logseq____",logseq____"~u652944b0-05f9-49be-af75-5c3e9b21343flogseq____",536870923]],[logseq____"^15logseq____",[236,logseq____"^Qlogseq____",logseq____"Für das leere Universum gilt\\n\\\\begin{align}\\n\\\\forall x: p(x)\\\\quadlogseq____&Tautologie\\\\\\\\\\n\\\\exists x: p(x)\\\\quadlogseq____&Kontradiktion\\\\\\\\\\n\\\\end{align}logseq____",536870923]],[logseq____"^15logseq____",[236,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[236,logseq____"^Flogseq____",229,536870923]],[logseq____"^15logseq____",[236,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[236,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[236,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[236,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[236,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[236,logseq____"^;logseq____",logseq____"~u652944b0-5963-4e49-b59e-c42524f7d168logseq____",536870923]],[logseq____"^15logseq____",[237,logseq____"^Qlogseq____",logseq____"[[Satz]] mit [[Variablen]] $x_1,\\\\cdots,x_n$, der bei Ersetzung jedes $x_j$ durch ein [Objekt](Objekt (Logik)) aus $U_j$ stets zu einer [Aussage](Aussagen) wirdlogseq____",536870923]],[logseq____"^15logseq____",[237,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[237,logseq____"^Flogseq____",240,536870923]],[logseq____"^15logseq____",[237,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[237,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[237,logseq____"^Ulogseq____",149,536870923]],[logseq____"^15logseq____",[237,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[237,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[237,logseq____"^Ulogseq____",207,536870923]],[logseq____"^15logseq____",[237,logseq____"^Hlogseq____",149,536870923]],[logseq____"^15logseq____",[237,logseq____"^Hlogseq____",207,536870923]],[logseq____"^15logseq____",[237,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[237,logseq____"^;logseq____",logseq____"~u652944b0-d035-47e9-9a46-10f0c47f79b0logseq____",536870923]],[logseq____"^15logseq____",[238,logseq____"^Qlogseq____",logseq____"see: [[assoziativ]] [[kommutativ]] [[idenpotent]] [[wechselseitig distributiv]] [[distributiv]] [[de Morgansche Regeln]]logseq____",536870923]],[logseq____"^15logseq____",[238,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[238,logseq____"^Flogseq____",217,536870923]],[logseq____"^15logseq____",[238,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[238,logseq____"^Vlogseq____",217,536870923]],[logseq____"^15logseq____",[238,logseq____"^Ulogseq____",190,536870923]],[logseq____"^15logseq____",[238,logseq____"^Ulogseq____",194,536870923]],[logseq____"^15logseq____",[238,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[238,logseq____"^Ulogseq____",196,536870923]],[logseq____"^15logseq____",[238,logseq____"^Ulogseq____",197,536870923]],[logseq____"^15logseq____",[238,logseq____"^Ulogseq____",202,536870923]],[logseq____"^15logseq____",[238,logseq____"^Ulogseq____",203,536870923]],[logseq____"^15logseq____",[238,logseq____"^Hlogseq____",190,536870923]],[logseq____"^15logseq____",[238,logseq____"^Hlogseq____",194,536870923]],[logseq____"^15logseq____",[238,logseq____"^Hlogseq____",196,536870923]],[logseq____"^15logseq____",[238,logseq____"^Hlogseq____",197,536870923]],[logseq____"^15logseq____",[238,logseq____"^Hlogseq____",202,536870923]],[logseq____"^15logseq____",[238,logseq____"^Hlogseq____",203,536870923]],[logseq____"^15logseq____",[238,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[238,logseq____"^;logseq____",logseq____"~u652944b0-00d2-4104-83d6-4188a4aeda69logseq____",536870923]],[logseq____"^15logseq____",[239,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____",[239,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[239,logseq____"^Flogseq____",227,536870923]],[logseq____"^15logseq____",[239,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[239,logseq____"^Vlogseq____",227,536870923]],[logseq____"^15logseq____",[239,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[239,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[239,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[239,logseq____"^;logseq____",logseq____"~u652944b0-9386-400a-881d-f06fc42c5c3alogseq____",536870923]],[logseq____"^15logseq____",[240,logseq____"^Qlogseq____",logseq____"[[Aussageform]] (AF) über [Universen](Universen (Logik)) $U_1,\\\\cdots,U_n$:logseq____",536870923]],[logseq____"^15logseq____",[240,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[240,logseq____"^Flogseq____",210,536870923]],[logseq____"^15logseq____",[240,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[240,logseq____"^Vlogseq____",210,536870923]],[logseq____"^15logseq____",[240,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[240,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[240,logseq____"^Hlogseq____",204,536870923]],[logseq____"^15logseq____",[240,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[240,logseq____"^;logseq____",logseq____"~u652944b0-e4ad-4633-8e9b-c1e48e3804d6logseq____",536870923]],[logseq____"^15logseq____",[241,logseq____"^Qlogseq____",logseq____"Schachtelung von quantoren ([Beispiel](Beispiele)):logseq____",536870923]],[logseq____"^15logseq____",[241,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[241,logseq____"^Flogseq____",228,536870923]],[logseq____"^15logseq____",[241,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[241,logseq____"^Vlogseq____",209,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",189,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[241,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[241,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[241,logseq____"^;logseq____",logseq____"~u652944b0-c8d9-4037-b035-9f3d2906ebadlogseq____",536870923]],[logseq____"^15logseq____",[242,logseq____"^Qlogseq____",logseq____"see: [[Große Azzoziativgesetze]] [[Große de Morgansche Regeln]] [[Negationsregeln]]logseq____",536870923]],[logseq____"^15logseq____",[242,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[242,logseq____"^Flogseq____",228,536870923]],[logseq____"^15logseq____",[242,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[242,logseq____"^Vlogseq____",228,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",189,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",191,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",201,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[242,logseq____"^Ulogseq____",206,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",191,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",201,536870923]],[logseq____"^15logseq____",[242,logseq____"^Hlogseq____",206,536870923]],[logseq____"^15logseq____",[242,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[242,logseq____"^;logseq____",logseq____"~u652944b0-fd16-4144-b35f-5012963bbb4flogseq____",536870923]],[logseq____"^15logseq____",[243,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist wahr}\\n\\\\end{align}\\n[Beweis](Beweise):logseq____",536870923]],[logseq____"^15logseq____",[243,logseq____"^Ologseq____",logseq____"^16logseq____",536870923]],[logseq____"^15logseq____",[243,logseq____"^Flogseq____",219,536870923]],[logseq____"^15logseq____",[243,logseq____"^Xlogseq____",195,536870923]],[logseq____"^15logseq____",[243,logseq____"^Vlogseq____",219,536870923]],[logseq____"^15logseq____",[243,logseq____"^Ulogseq____",195,536870923]],[logseq____"^15logseq____",[243,logseq____"^Ulogseq____",198,536870923]],[logseq____"^15logseq____",[243,logseq____"^Ulogseq____",204,536870923]],[logseq____"^15logseq____",[243,logseq____"^17logseq____",true,536870923]],[logseq____"^15logseq____",[243,logseq____"^;logseq____",logseq____"~u652944b0-589d-431e-a1bc-197fe619bc32logseq____",536870923]],[logseq____"^15logseq____",[244,logseq____"^Qlogseq____",logseq____"\\\\begin{align}\\n\\\\forall x:\\\\exists y:xlogseq____<y\\\\quad\\\\text{ist 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