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Set TheoryQuestion and Answers: Page 1

Question Number 204141 Answers: 1 Comments: 0

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$$\:\:\: \\ $$

Question Number 203490 Answers: 1 Comments: 0

1×3×5×7×9×...×2005 = ... (mod 1000)

$$\:\:\:\:\mathrm{1}×\mathrm{3}×\mathrm{5}×\mathrm{7}×\mathrm{9}×...×\mathrm{2005}\:=\:...\:\left(\mathrm{mod}\:\mathrm{1000}\right) \\ $$

Question Number 199624 Answers: 0 Comments: 1

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Question Number 196950 Answers: 1 Comments: 0

Prove that ∫^( (π/2)) _( 0) ((ln(1+αsint))/(sint))dt= (π^2 /8)−(1/2)(arccosα)^2

$$\mathrm{Prove}\:\mathrm{that}\:\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{1}+\alpha\mathrm{sin}{t}\right)}{\mathrm{sin}{t}}{dt}=\:\frac{\pi^{\mathrm{2}} }{\mathrm{8}}−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{arccos}\alpha\right)^{\mathrm{2}} \\ $$

Question Number 195393 Answers: 1 Comments: 0

prove that lim_(x→0) (((Σ_(k=1) ^n (1−(1/(2k)))^x )/n))^(1/( x )) = (1/4)(C_(2n) ^n )^(1/n)

$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt[{\:\:\boldsymbol{{x}}\:\:}]{\frac{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}{k}}\right)^{{x}} }{{n}}}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\sqrt[{\boldsymbol{{n}}}]{\mathrm{C}_{\mathrm{2}\boldsymbol{\mathrm{n}}} ^{\boldsymbol{\mathrm{n}}} } \\ $$

Question Number 200284 Answers: 1 Comments: 0

Prove that for any set A containing n elements, ∣P(A)∣=2^n .

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{set}\:{A}\:\mathrm{containing}\:{n} \\ $$$$\mathrm{elements},\:\mid\mathcal{P}\left({A}\right)\mid=\mathrm{2}^{{n}} . \\ $$

Question Number 195157 Answers: 1 Comments: 0

Prove that (x^3 /(2sin^2 ((1/2)arctan (x/y))))+(y^3 /(2cos^2 ((1/2)arctan (y/x))))=(x+y)(x^2 +y^2 )

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\frac{{x}^{\mathrm{3}} }{\mathrm{2}{sin}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}{arctan}\:\frac{{x}}{{y}}\right)}+\frac{{y}^{\mathrm{3}} }{\mathrm{2}{cos}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\mathrm{2}}{arctan}\:\frac{{y}}{{x}}\right)}=\left({x}+{y}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$

Question Number 194868 Answers: 1 Comments: 0

Prove that ∀n∈IN ∫^( 1) _( 0) t sin^(2n) (lnt)dt= (1/(1−e^(−2π) )) ∫^( π) _( 0) e^(−2t) sin^(2n) (t)dt

$$\mathrm{Prove}\:\mathrm{that}\:\forall{n}\in\mathrm{IN} \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} {t}\:{sin}^{\mathrm{2}{n}} \left({lnt}\right){dt}=\:\frac{\mathrm{1}}{\mathrm{1}−{e}^{−\mathrm{2}\pi} }\:\underset{\:\mathrm{0}} {\int}^{\:\pi} {e}^{−\mathrm{2}{t}} {sin}^{\mathrm{2}{n}} \left({t}\right){dt} \\ $$

Question Number 194781 Answers: 0 Comments: 0

f_(n ) the general sentence is seqiencee fibonacci. prove that : f_(2n−1) =f_n ^2 +f_(n−1) ^2

$${f}_{{n}\:} \:\:{the}\:{general}\:{sentence}\:{is}\:{seqiencee} \\ $$$${fibonacci}.\: \\ $$$${prove}\:{that}\::\:\:{f}_{\mathrm{2}{n}−\mathrm{1}} ={f}_{{n}} ^{\mathrm{2}} +{f}_{{n}−\mathrm{1}} ^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 194709 Answers: 1 Comments: 0

Show that in fibonacci sequence f_(3n) =f_n ^3 +f_(n+1) ^3 −f_(n−1) ^3

$${Show}\:{that}\:\:{in}\:{fibonacci}\:{sequence} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{f}_{\mathrm{3}{n}} ={f}_{{n}} ^{\mathrm{3}} +{f}_{{n}+\mathrm{1}} ^{\mathrm{3}} −{f}_{{n}−\mathrm{1}} ^{\mathrm{3}} \\ $$$$ \\ $$

Question Number 194693 Answers: 1 Comments: 0

if f_n =f_(n−1) +f_(n−2) ; f_1 =f_2 =1 then prove that 5∣f_(5n)

$${if}\:\:\:{f}_{{n}} ={f}_{{n}−\mathrm{1}} +{f}_{{n}−\mathrm{2}} \:\:;\:\:{f}_{\mathrm{1}} ={f}_{\mathrm{2}} =\mathrm{1} \\ $$$${then}\:\:\:{prove}\:{that}\:\:\:\mathrm{5}\mid{f}_{\mathrm{5}{n}} \:\: \\ $$

Question Number 194446 Answers: 0 Comments: 0

Question Number 191731 Answers: 1 Comments: 0

Use laws of algebra to prove the following (a)[(B−A)u(A−B)]=[(AuB)−(AnB)] (b)A▽(AnB)=A−B

$$\mathrm{Use}\:\mathrm{laws}\:\mathrm{of}\:\mathrm{algebra}\:\mathrm{to}\:\mathrm{prove}\:\mathrm{the}\:\mathrm{following} \\ $$$$\left(\mathrm{a}\right)\left[\left(\mathrm{B}−\mathrm{A}\right)\mathrm{u}\left(\mathrm{A}−\mathrm{B}\right)\right]=\left[\left(\mathrm{AuB}\right)−\left(\mathrm{AnB}\right)\right] \\ $$$$\left(\mathrm{b}\right)\mathrm{A}\bigtriangledown\left(\mathrm{AnB}\right)=\mathrm{A}−\mathrm{B} \\ $$

Question Number 191501 Answers: 1 Comments: 2

find a solution; e^x = ln(x)

$$ \\ $$$$\:\:\:\:{find}\:{a}\:{solution}; \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{e}}^{\boldsymbol{{x}}} \:=\:\boldsymbol{{ln}}\left(\boldsymbol{{x}}\right) \\ $$$$ \\ $$

Question Number 190573 Answers: 1 Comments: 3

a^ 2+2ab+b^ 2

$$\hat {{a}}\mathrm{2}+\mathrm{2}{ab}+\hat {{b}}\mathrm{2} \\ $$

Question Number 189672 Answers: 0 Comments: 2

Question Number 189091 Answers: 1 Comments: 0

1 : Ω = Σ_(n=1) ^∞ (( (− 1 )^( n) H_( n) )/n^( 2) ) = ? 2 : η (−1 )= ?

$$ \\ $$$$\:\:\:\:\mathrm{1}\::\:\:\:\:\Omega\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(−\:\mathrm{1}\:\right)^{\:{n}} \mathrm{H}_{\:{n}} }{{n}^{\:\mathrm{2}} }\:=\:? \\ $$$$\:\:\:\:\mathrm{2}\::\:\:\:\:\:\eta\:\left(−\mathrm{1}\:\right)=\:? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 185890 Answers: 0 Comments: 0

Q: G( V , E ) is a graph , such that ∣ V (G )∣ = 20 , Δ ( G )= 8 , δ (G )=3 find the value of , q_( max) − q_( min) = ? q = ∣ E (G )∣

$$ \\ $$$$\:\:{Q}:\:\:\:\:\:{G}\left(\:{V}\:,\:{E}\:\right)\:\:{is}\:{a}\:{graph}\:,\:{such}\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mid\:\:{V}\:\left({G}\:\right)\mid\:=\:\mathrm{20}\:,\:\:\Delta\:\left(\:{G}\:\right)=\:\mathrm{8}\:,\:\:\delta\:\left({G}\:\right)=\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{find}\:\:{the}\:{value}\:{of}\:\:,\:\:{q}_{\:{max}} \:−\:{q}_{\:{min}} \:=\:? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{q}\:=\:\mid\:{E}\:\left({G}\:\right)\mid\:\: \\ $$

Question Number 182718 Answers: 2 Comments: 4

If , 7^( n) ≡^(10) 7^( 19) then find the 1st digit of the numer , 8^( n+4) .

$$ \\ $$$$\:\:\:\:\:\:\mathrm{If}\:,\:\:\:\:\mathrm{7}^{\:{n}} \:\overset{\mathrm{10}} {\equiv}\:\mathrm{7}^{\:\mathrm{19}} \\ $$$$\:\:\:\:\:\:\:{then}\:\:{find}\:{the}\:\:\mathrm{1}{st}\:{digit} \\ $$$$\:\:\:\:{of}\:\:{the}\:{numer}\:\:,\:\:\:\mathrm{8}^{\:{n}+\mathrm{4}} \:.\: \\ $$$$\:\:\:\:\:\:\:\: \\ $$

Question Number 182203 Answers: 0 Comments: 0

Let A={1^(p^2 −p) , 2^(p^2 −p) ,..., (p−1)^(p^2 −p) , p^2 −p+1} where p is any prime number Prove that for any value of p, however we split this set into two disjunctive sets, the arithmetic means of all elements of both sets cannot be equal to each other.

$${Let}\:{A}=\left\{\mathrm{1}^{{p}^{\mathrm{2}} −{p}} ,\:\mathrm{2}^{{p}^{\mathrm{2}} −{p}} ,...,\:\left({p}−\mathrm{1}\right)^{{p}^{\mathrm{2}} −{p}} ,\:{p}^{\mathrm{2}} −{p}+\mathrm{1}\right\} \\ $$$${where}\:{p}\:{is}\:{any}\:{prime}\:{number} \\ $$$${Prove}\:{that}\:{for}\:{any}\:{value}\:{of}\:{p}, \\ $$$${however}\:{we}\:{split}\:{this}\:{set}\:{into}\:{two} \\ $$$${disjunctive}\:{sets},\:{the}\:{arithmetic} \\ $$$${means}\:{of}\:{all}\:{elements}\:{of}\:{both}\:{sets} \\ $$$${cannot}\:{be}\:{equal}\:{to}\:{each}\:{other}. \\ $$

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