6. Prove that if alb and k E Z, then a|(b+ ak).

Q: - 21 , Prove that + (n? m' ) yn=o 2.

A: According to the Leibnitz theorem, dndxn(uv)=C0nunv0+C1nun-1v1+C2nun-2v2+... Formulae used:…

Q: Given that en 2= V and en 3=W,express each of the fallowing in terms of vand w a) in zv0/3

A:

Q: If a+a e c(R) for all ae R. then Commutative is prove that

A:

Q: * If Iz| = Re(z), then z is

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 by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

A:

Q: Let (W, t > 0) be standard BM. Show that P(n|w,1 – Wa sc) → as n → 0.

A: From the given information,  (Wt, t≥0) be standard Brownian Motion. Note: In the given situation,…

Q: prove that: DO e de a e +6 2/3 cwhere a, b > o

A: Where a, b>0

Q: HeW. By the Same way Prove that: X Ju (x) (X)

A:

Q: Prove: If = ¿ (where a, b, c, and d are nonzero), b then d*

A: Given:               ab=cd  (where a,b,c  and d are non zero.)

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

A:

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

A:

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: d set): Let (M ,d) be a (SUT)' = S' U T'.

A:

Q: 3. Show that the function p : F → Ę which maps o to o is a bijection from F to Fe, where and ar to…

A: Let ϕ:F1→F2 be a function such that, ϕ0=0. Also given that ϕαr=β3r, where 0≤r≤6 When r=0,…

Q: 5. Prove that if A and B are similar then dim(ker(A)) = dim(ker(B)).

A: We have to prove dim(ker(A)) = dim(ker(B)).

Q: Let X ~ N(0, 1). For any € > 0, show that P(|X| > c) ≤ ²c - ??

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

A:

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: 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: Suppose T: R + P, has an action on a

A: Given, And

Q: verify that (AT )T = A

A:

Q: Prove that Z(Sn) = {e} for n ≥ 3.

A:

Q: Shaa thut the graf of lincs iju fins rom F from E Iwhore is a iomasphic wilh Ge(5)

A:

Q: Let q E Q and x E R - Q. Prove that q + XER - Q.

A:

Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: discrete

A: In method of contradiction we start by assuming that given statement is false. Then prove that it is…

Q: Show that if x is in both W and W-, then x = 0.

A: Dot product of perpendicular vectors is 0.

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: If z, w E C, prove that z+w=z+w.

A: This follows from simple substitution. The detailed solution is presented in step 2. As requested i…

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

A:

Q: c If in is an involution of X, show that (x, y)in = (yin, xin) is an involution of X2). %3D

A: Let X be a lattice We have to show that (x, y)in=(yin, xin) is an X[2] X[2] is a 2 x 2 matrix We…

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

A:

Q: 2) Let a,bE Q such that a<6. Prove that ax <bx.

A: The statements can be proved by analyzing the results from the given statements. The other methods…

Q: 3. Show that X = (x|Ax=b, x ≥ 0} is convex. following ure which no

A: As per our guidelines we are supposed to answer only one question.

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

A:

Q: Suppose A · B = 0. Show that this implies that C(B) is contained in Null(A).

A: Suppose A.B = 0 then we have to show that  Column space of B contained in the null space of A  i.e…

Q: 4.2. Let a, b E Z, where a 0 and b 0. Prove that if alb and bla, then a = b or a = -b. -b.

A:

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

A:

Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: c) Compute µr, Hy, Ož, o, Cov(X, Y), and p

A: Reviewing information, Given pdf is, f(x,y)=x+y                 ; 0≤x≤1 ,  0≤y≤1

Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

A:

Q: Let S = {(x, y) :x² + (y – 1)² < 1} U {(3,0)}. %3D Is the origin an extreme point of conv S? Is the…

A:

Q: In Z, let H = &lt;5&gt; and K = &lt;7&gt;. Prove that Z = HK. Does Z = H X K?

A: Given: In Z H=5K=7

Q: Show that Oa = {a} if and only if a e Z(G).

A: For a∈G, Oa denote the orbit of 'a' under the conjuction action.The orbits under the conjucation…

Q: Suppose that B is a point between A and C. Prove that AB−→ =AC−→

A: B is a point on AC. Let, A=(x1,y1) and C=(x3,y3)Then direction vector of AC=(x3 - x1, y3-y1)

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

A: The complete solutions are given below