Let R be and bl#0) = iff a with unity - Let a l#0) in R- Show that and b an I.D are are associates-

Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: Let x, y e Z. Prove that if 3x+7y is odd or 5x + 6y is even, then x or y is even.

A: We have given that for x,y belongs to integers. If 3x+7y is odd or 5x+6y is even then x or y is even…

Q: Let x, y, z E R. Prove that if x – 2y + 3z is odd, then 4xy – 6z is even.

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

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Q: Z1 is complementary to ZM 22 is complementary to ZM Z1 = 2 Given: Prove: M R 2

A: Given: ∠1 is complimentary to ∠M. ∠2 is complimentary to ∠M. To prove: ∠1≅∠2 According to the…

Q: let H = {n² |n e Z*}, determine whether it is closed under

A: Here, we have H=n2| n∈ℤ+. To show the set is closed under binary multiplication. This means we have…

Q: 1d) Prove: If z=Zo is a zero of order n of flz), then it is zero of order 2n of

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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 Hence deduce that L{x"; p} r(n + 1) pn+1 " if n>-1, p > 0. L{x-¹/2; p} = √√(π/p),p> 0

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Q: Prove or give an counterexample to the statement that for any neZ+ and aeZn, the equation +n X = a…

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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: (a) Let a > 0. Using the substitution t = In x or otherwise, prove that In a e 4/a² - e 2t dt =a5…

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Q: Show that the finite difference formula 1 Dzu(î) = [Зи (т) — 4u(г — h) + u(т — 2h)] 2h is a…

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Q: Let f(2)=³ and let C = {z € C : |z| = 1}, described in the positive sense. Determine the value of Ac…

A: Given that             fz=z4-5z2      and    C=z∈ℂ : z=1.

Q: -2 Let f(t,y) = y + 3t?e'. Then f satisfies a Lipschitz condition in y with L = 2, given that 1<t<2.…

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Q: J prove that for 0 <r<R

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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: (a) Let n €N\{0}, and k e Z. Prove that k-

A: We will use very basic approach for this question in such a manner that it will enter in your head…

Q: Suppose I is an open interval and f € C°(I). Show by induction that ffor х, а € 1, f(x) = f(a) +…

A: Given: I is an open interval and f∈C∞I   To show: fx=fa+∑k=1nfkak!x-ak+1n!∫axx-tnfn+1tdt for x,a∈I…

Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

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Q: Assume that |Xn] 0. (Hint: first, prove that P(|X|>c+1) = 0.)

A: Answer:

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

A: 3).

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: Let a,b E R with a # 0, b # 0. Prove that (ab)¯1 = a°1b´1. %3D

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

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Q: Consider the following integral: +oo J(2a) = | - dzerp(-20r") / dyezp(-2ay7) = { / drerp(-2aa")}?…

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Q: Evaluate z'dz where C is z=2 by Residue theorem. (z-1) (z-3) (s-2) ,(1-2)}

A: Given A complex integral we can use cauchy's residue theorem to solve the problem

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: Let a = 8 and b = 9. Find m,n E Z such that am+bn =1.

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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: omplete his praol of Theoren 1. B Cive ZA and ZB with AĎ || BẺ; AČ || BĚ. F D Prove: ZA is…

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Q: a. Show that T: 1² → 1² defined by Tx = y = (n), n, = r, is compact where x = (5,) and where0 <r< 1.

A: Show that T: l2→l2 defined by Tx=y=ηj, ηj=rjξj is compact where x=ξj and 0&lt;r&lt;1. Definition:…

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: Suppose m, n ∈ Z + are relatively prime. The function f : Zmn → Zm × Zn defined by  f([Z]mn) =…

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Q: Let r E Rt. Prove that if z – 2x2 +2 2021. -

A: Let x2=t, Now if x∈R+⇒t∈R+ ⇒t2-2t+2≤0⇒t-12+1≤0⇒t-12≤-1which is not possible for any value of t, for…

Q: Given the function z = In (2xy + e3x) - 5y, %3D az a. ax

A: Since, we have to find the partial derivatives ∂z∂x and ∂z∂y of functiongiven by z=ln…

Q: Let z E C. Prove that if 1+z +...+z"-1 – nz" = 0, then |2| <1.

A: Given that, z∈C and, 1+z+...+zn-1-nzn=0 It is known that, 1+z+z2+z3+....+zn-1=1(rn-2-1)(r-1) So the…

Q: Please solve...

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Q: Let A e Mnxn(F) be upper triangular. Prove that det(A) is the product of the diagonal entries.

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Q: (a) Compute /(2) dr, given that f(b) 3 and that | af'(a) dr b(1 + f(-b)). (b) Computei (2) dz. given…

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Q: B. Prove by direct proof general. P: Vn e ZE, (n – 1) (n+ 1) e Z,

A: I have solved B; you just need to expand (n-1)(n+1) and then use the fact that square is always…

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: Let n E Z. Show that f(x) = x3 – nx +2 is irreducible over Q for %3D n + -1,3, 5.

A: We shall find all n ∈ Z such that f(x) is reducible. Assume that f(x) is reducible. As f(x) is…

Q: In C[0, 27], with the inner product -2 (f, 8) = | f(x)g(x) dx, show that the set S = {1, sin x, cos…

A: Given: Inner product defined as: &lt;f,g&gt; = ∫02πf(x)g(x)dx To show that the set S = { 1, sinx,…

Q: Find conditions on a, b, c, and d such that

A: Find conditions on a, b, c, and d such that  B=abcd commutes with both 1000 and 0001 For any matrix…

Q: Prove that e“ p.w. on R. n! n=0 M8

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Q: Find the value(s) of k such that A is singular.

A: to find the value of k

Q: Let x, y e Z. Prove that if 3x + 7y is odd or 5x + 6y is even, then x or y is even.

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.