Prove that for any k > 0, nk e o(2v™).

Q: Prove that if a, b e Z – {0}, then gcd(a, b) = gcd(a, -b)

A: Solution:-   gcd a,b= Greatest common divisor of  a  and  b ∗Every integer  that divides  a  and  b…

Q: Prove by mathematical induction that, for all positive integers n, de (e* sin x) = 23"e* sin(x…

A: To Prove →dndxn ex sin x = 212n ex sinx + 14nλBy mathematical induction for all positive integen…

Q: Prove that det(At) = det(A) for any A ∈ M2× 2(F).

A: Let us consider a 2*2 matrix A whose entry belongs to F, which is given below

Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

A:

Q: Suppose that > ak = a and let c E R. Prove, in all the that (c-an) = gory detail necessary, = c•Q.…

A: We will give two proofs one based on partial sums and the other will be a direct proof.

Q: Prove that tr(aA + bB) = atr(A) + b tr(B) for any A, B belongs to Mn× n(F).

A: We need to prove that tr(a A + b B) = a tr(A) + b tr(B) for any A, B belongs to Mn×n(F).

Q: Prove that S4 is not isomorphic to D12.

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Q: Let A € M₂ (R) be given. (a) If AA¹ = 0n, prove that A = On. Does this hold if A € M₂(C)? (b)…

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Q: By using caushy Rieman prove that ·C. Prove thatd az az e : ae

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Q: Prove for all mEw and n E w, we have that m+ ·n=m・n+n.

A: Given that m,n∈ω. To prove: m+· n=m·n+n

Q: Prove that if AT A = A, then A is symmetric and A = A².

A:

Q: Given that n ≥ 2 , x ∈ Rn  Prove that there exists a y ∈ Rn in where y ≠ 0, and xy = 0

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Q: If the Wronskian W of f and g is 3e^(4t), and f(t) = e^(2t), find g(t)

A: Wronskian W of f and g is given by:

Q: Prove that for all positive integers n.

A: To prove for all positive integer n ∫01(lnx)ndx=(-1)nn!

Q: 3. Use the above Lemma and Mathematical Induction to generalize the previous exercise. That is,…

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Q: prove that erf (x) dx =o

A: To prove: ∫−aaerfxdx=0

Q: In the domain of Gaussian integers Zli), the element 19 is * O Reducible O Unit Irreducible

A: Answer is irreducible. Proof is given below.

Q: Suppose that C> 1. Prove that if t > Vm ln C+In C, then ()/F) >C.

A: We will use the following Strining's approximation :  n! ≈ √(2πn) e-n nn To prove the given…

Q: Let x E Q'. Show that for any n EN, there exists Xn E Q such that |x- Xal <

A: Given x∈ℚ', that x is a fixed irrational number. We know that between any two real  number, there…

Q: U prove that In (l+x)) 2.

A:

Q: Let h : N x N → N be the function h(x, y) = 3x + 2y. Use strong induction to prove that if n EN and…

A: Given that h : ℕ×ℕ→ℕ be the function h(x,y)=3x+2y. Then we have to prove by using strong induction…

Q: Prove that for each n E N, (d) (n) e (d) dn

A: To prove, for each n∈ℕ, nφn=∑d|nμ2dφd. We know that the mobius function μn is defined as: μn=1if…

Q: 3. Let n E Z. Give a direct proof that there exist x, y E Z such that n = 55x + 78y.

A: Objective : To prove for any n∈ℤ ∃x,y∈ℤ such that n=55x+78y Prove for n=1, Greatest common divisor…

Q: Prove that for all values of the constant A, eAInt = e In.

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Q: nd C such that AC=E

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Q: = Prove that E = ehd.

A: We know that Ef(x)=f(x+h)

Q: If A is similar to B, prove that AT is similar to BT.

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Q: Let A E Mnxn (F) and B = aA, with a E F. Prove by induction that det(B) all positive integer n. a"…

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Q: Let x, y, A, u E Z. Prove that if 1 = Ax + µy then r, y are relatively prime.

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Q: If e denotes the unity element in an integral domain D, prove that me · ne = (mn)e for all т, п € Z

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Q: Prove that if a, b E Z - {0}, then gcd(a, b) = gcd(a, -b)

A: soln:gcda,b=Greatest common divisor of a and bEvery integer that divides a and bsimultaneously will…

Q: Suppose a E Z. Prove that for any n E N, 5|2"a implies 5|a.

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Q: If e is the unity in an integral domain D , prove that (−e)a = −a for a D .

A:

Q: Show that for any positive integers i and j with i > j, we have T;(x)T;(x) = [Ti+;(x) +T;-j(x)].

A: we have to show, for any positive integers i and j with i&gt;j TixTjx=12Ti+jx+Ti-jx Now, we know…

Q: 3. Find the generator of X = Xo + ut + oWt, where W is a standard BM, and identify the PDE satisfied…

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Q: If e denotes the unity element in an integral domain D, prove that me • ne = ( mn )e for all m, n ϵ…

A: We prove the identity as,

Q: Let ECR be measurable. Prove that L'(E)n L°(E) C L*(E).

A: Given E⊆ℝ be a measurable set. Let f∈L7E∩L9E. Let λ=716, therefore:…

Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: Let P = {X E P(Z*)|X is finite}. Prove that P is denumerable.

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Q: - In the domain of Gaussian integers Z[i], the element 43 is O irreducible O reducible O unit

A:

Q: For any x E R, there exists n EN such that x < n; ylgn ihi

A: We have to state whether given statement is true or false.

Q: Let A E Mnxn(F) and B = aA, with a E F. Prove by induction that det(B) = a" det(A), for all positive…

A:

Q: Let a > 1 be an integer. Prove that n|p(a" – 1), where o is the Euler ç-function.

A: In number theory, Euler's totient function counts the positive integers up to a given integer n that…

Q: Prove that : a"+ (a+ 1)° =1 +a(1+い) (mod a la+s) for all a, m,n E Z.

A:

Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: Let x, y,p E Z, where p is prime. Prove that x2 – p²y – p + 0. (Hint: you may use without proof that…

A: Solution :-

Q: Prove that det(kA) = kn det(A) for any A ∈ Mn×n(F).

A: In determinant the scalar quantity is taken common from a single row or a column.