1. Let A = {x E Z | 6 divides x} and B = {x € Z | 3 divides x}. Prove that A C B, and then prove that B Į A.

Q: 1: Prove that (c a) * H = (c b) * H implies that a H = b H

A: Given an equation. We need to prove that : a*H = b*H where H is a subgroup of a group ( say G). Also…

Q: If z| =5 and w =- z+5 then Re(w) is equal to

A: Introduction :

Q: Calculate M, for f(x) = 6 · In(æ³) over [1, 2]. M6 ||

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Q: If u (E) =0, then E is measurable.

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

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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 x, y E R. Prove that if 21x + 35y = 45, then x and y are not both integers.

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Q: Given: 23 = Z1 Prove: АВ АС 2 3/6 E

A: Given: The angle ∠3 is equal to ∠1. Concept used: If the two angle of a triangle is equal then side…

Q: Show that For all z E C, with z # 0, then: |z/²z=1 = z

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Q: Let A = {x ∈ Z | 6 divides x} and B = {x ∈ Z | 3 divides x}. Prove that A ⊆ B, and then prove that…

A: We can prove using like showing every element in one set is an element of another set like that.

Q: For any x ER, prove that there exists q E Z such that q-1<r <q.

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Q: b) If d = 1, prove that (1 if a is odd gcd(a, 2b) %3D if a is even

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Q: If m and n are positive integers and w = g(x"y"), then show that dw dw = my dy

A: Given w=g(xmyn).

Q: Show that if ax2 + bx + c = 0 for all x, then a = b = c =0

A: It is given that, ax2+bx+c=0 for all x,  To prove a = b = c = 0

Q: Classify e + 3y = x²y sexactly one of the following: as

A: We have to classify the given differential equation

Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

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Q: Let X and Y be defined as follows: X = {a E Z | a = 3x for some x E Z}, and Y = {b E Z |b = 3y – 3…

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Q: Prove that if A, B ϵ Mnxn(F) are similar, then det{A) = det(B).

A: Given:  A, B ∈ MnxnF are similar To prove:  det(A)=det(B).

Q: Let P = {p:p=6x-1, for some x EZ} and Q= {q:6y+5, for some y EZ). Prove that P= Q

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Q: Suppose: n ≥ 3 and x,y ∈ Rn , |x - y| = d > 0 and r is some +R, Need to show: 2r = d, there is…

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Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: 5. Prove that if x, y ER, then x + y ≤ x + y. Use the following cases: (i.) x, y ≥ 0, (ii.) x, y <…

A: We now prove that if x,y ∈ ℝ , then x+ y≤ x+y with the help of given three cases.

Q: Show that for n, m e Z+, gcd(2" – 1,2m – 1) = 2gcd(n,m) – 1.

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Q: Prove: (i) b (A) CA if and only if A is closed. (ii) b(A) n A 9 if and only if A is open. (li1) b(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: 2. (Z, * ) where * is define by a * b = a - b.

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Q: If A is a 2 x 2 with det(A) = 3, then det(4A1) = %3D

A: det(A-1)=1det(A) det(4A-1)=4ndet(A-1) where n is the order of the matrix.

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: 21. Find af and of if f(x,4)= 3x" +2xy+ Sy , x= 4v? + Bn", and u 2v-w? Vour %3D 4= 2v-w? Vour av…

A: We have to check

Q: a) By using the substitution u = √√x +3, evaluate S. 2 √(√₁ + 3)2 d. dr.

A: Given, ∫142xx+32dx

Q: Supposernc N and n 4 0 Then show that rn is positive if and only if r is positive

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Q: Show that if X gamma(r, A) and Y ~ gamma(s, ) independent, are then X + Y is gamma(r + s, ).

A: X~Gammar,λ fx=λrΓre-λxxr-1,     x&gt;0, r&gt;0, λ&gt;0 Y~Gammas,λ fy=λsΓse-λyys-1,     y&gt;0,…

Q: show that if Z ~ N(0, 1) and Y ~x²(n) with Z and Y to be independent, then T: n

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Q: If m` E = 0, then E is measurable.

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Q: If C = {x:x² + 11x + 30 = 0, x ∈ N}, then C is a/an ?

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Q: 53. Suppose that a + 0. (a) If a · b = a·c, does it follow that b (b) If a X b = a X c, does it…

A: Given a, b, c are three vectors and a ≠ 0.

Q: es lls ew zeiflucifiib ons einiseen stigesber Show that for neN, (2n)! ' (r!)² ((n −r)!)² - 7=0 2n -…

A: We will use binomial expansion to prove given relation

Q: t, for t > 0, sa X 2 is

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Q: 6. Prove that, if a, b e Z, then 3a? – 4b – 5 0. - -

A: Answer

Q: Suppose X~ N(u=120, o = 3), determine the value of x such that P(X>x) = 0.05.

A: Given that X follows N(mean = 120, SD = 3), then Z = (X - mean)/SD follows N(0,1)

Q: let X~~ Per(P) Prove ô is unbiare if ô-X let x exp(0) Prave e is unbiar

A: Introduction: One random variable follows Bernoulli with parameter p and other random variable…

Q: In P2 , if B = {1,1+x+x², x + 2} and D = {x²,x, 1} , then Pp-B = 1 2 0 1 1 1 1 0 1 0 1 1 1 1 2

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Q: Show that if (B) (C) = 3x N (AB)_ (BC) (CA) =y, N (A) = x, = 2x, and N N N then the value of neither…

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Q: If det A= 0 why is det A+ = O? If A has rank r, why does A+ have rank r?

A: Given Data det A=0   Suppose the column and row of the matrix are linearly independent. The matrix…

Q: x | X> 3) =? and E [X | X> 3] =?

A: X ∼ exp (λ)  having pdf  f(x)=λe-λx      x&lt;0

Q: By using Caushy Rieman prove that (2) and (3) (2) 4ca) = e (3) .w prove Thot()-

A: (a)

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

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Q: Let P= (p:p=6x-1, for some x EZ) and Q = {q:q=6y+5, for some y EZ}. Prove that P=Q

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