Let S and T be subset of R. Prove the following: cl(SUT) = (cl S) U (cl T)

Q: Complete the following proof. Given: FG FJ, HG = IJ Prove: HF = IF

A: I have provided solution in step2.

Q: Let J={a: aER}, then I is Borel algebra.

A: False Statement' Detailed Explanation

Q: We defined NCE) = ZE for ZE Z Ii]. Prove N (Zw) = NLZ) Ncw) for all Z,W E Z6] .

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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: Prove that if Xn X and Ync, then (Xn, Yn) → (X, c).

A: Please see the answer innext step

Q: 1. Show that S, has the presentation (x, y|x² = y = (xy)² = 1). %3D

A: The symmetric group S3 is given as S3=I, 1 2, 1 3, 2 3, 1 2 3 1 3 2 where  I=123123, 1 2=123213, 1…

Q: If sup(S) exists, then it is unique ylgn ihi

A: The solution is given as

Q: Let r = xi + yj + zk and let r = ||r||. Prove that

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

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 the following relationship is true: nCr + nCr + 1 = n + 1Cr + 1

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Q: Let m eN and a E Z be such that (a, m) = 1. Let i, j e N. Prove that if om(a) = ij, then om(a') = j.

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

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Q: Given that w = f (u, v) And u = xz. , V = dw = X az aw aw Prove that Z - ду

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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: Let a, b, c E Z. Prove that if a b and alc, then Vk EZ we have a (b+ ck).

A: ab  &  ac   Since ab, therefore b=a·m   ,  where m∈ℤ Since ac, therefore b=a·n   ,  where n∈ℤ

Q: Let A, Be Mnxn (F). (a) Prove that if A~ B, then At ~ Bt.

A: From the given information. The matrices A and B belong to M of order nxn. It is required to prove .

Q: Prove that TIN) F(a – p) O< Re

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Q: If S is a subset of T , then u* (T) su* (S).

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Q: Let c E R and suppose that an → a. Show that can → ca.

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Q: Let x, y E Z. Prove that if 3 ł x and 3 ł y, then 3|(2² – v²).

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Q: 10. Let X , j= 1,.., n, be i.i.d. r.v.'s such that E(X,)= µ, o'(X)=o', both finite. j' Show that E(X…

A: The normal distribution: A continuous random variable X is said to follow normal distribution if the…

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: Qt Let T= {carb) a,b ER\fopp and of defined onI by Carb)*(c,d)= (ac, bd) for all (a,b), (c,d ET,…

A: The set T is defined as, T=a, b | a, b∈ℝ\ 0 The operation * is defined on T as follows, a, b*c,…

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: Find f, and f, and evaluate each at the given point. f(x, y) = eXy2, (In 5, 2) fo(x, y) f,(x, v) =…

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Q: Let f(x, y, z) = x³ – yz?. Find af af of and ax' dy az

A: Given : fx,y,z = x3-yz2

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

A:

Q: ), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o

A: *Answer:

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 P = {(z, y, 2) E R° | az + by + cz = 0} where (a, b, c) # (0,0,0). (a) Prove that (a, b, c) ¢ P.…

A: Given that P=x,y,z∈ℝ3 : ax+by+cz=0 where a,b,c≠0,0,0 (a) The objective is to prove that a,b,c∉P Let,…

Q: If (a + bx)e = x, prove that x’y, =(xy,-y)

A: Since you have asked multiple questions , we will solve first question for you. If you want any…

Q: let it be T=11,2,3 } S:la,b,c] Proved P(TNS)E P(S)NP(T)

A:

Q: Use the diagram to prove that ACAS ATAS. A T. Given: CS = ST and mzCAS mzTAS= 90 Prove: ACAS= ATAS

A: Given : CS ≅ STm∠CAS = m∠TAS = 90°

Q: Prove the following existence proof. P: 3a, y, z E N such that ryz = 24

A: Existence proof

Q: If S is a subset of T, then u*(T) su* (S)

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Q: Let z, w e C. Clearly prove that z+w = z + w.

A: Since you have asked multiple question , we will solve the first question for you. If you want any…

Q: Show that (t As) can be derived from the premises pq, q→~r,r, pV (t As).

A:

Q: 1. Prove with the usual notations that: (b) A + V =- (a) EV = VE = A (c) A= SEZ

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that: If f E Ra, b), then |fI E Rla, b) with

A:

Q: a) Suppose that you wish to prove that ARST = ASRV. Using the reason Identity, name one pair of…

A: Given: We wish to have the triangle RST and triangle RSV to be congruent.

Q: Let R be a UFD and fx) in REX) So, 0+a € IR.. Show that claf)~ ac(f) a nonzero polynamial

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Q: Let X, Y, Z C U. If Pr(X) 0.19, Pr(Y) 0.33, Pr(Z) 0.45, Pr(Xn Y) 0.07, Pr(Xn Z) Pr(Y n Z) 0.16, and…

A: Given DataP(X) = 0.19P(Y) = 0.33P(Z) = 0.45

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

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Q: Let m e N and a, b e Z be such that (a, m) = 1 and (b, m) %3D = 1. Let h = 0m(a), k = 0m (b), and…

A: Let m∈ℕ and a, b∈ℤ be such that a, m=1 and b, m=1. Let h=oma, k=omb,  and  n=omab. We need to show…

Q: Show that for all x, y E F, there exists e with e < Emachine such that fl(x * y) = (x * y)(1+ €) is…

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Q: . Prove with the usual notations that: (b) A +V =- (c) A= SEŽ (a) EV = VE = A

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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: Suppose that for every r E R, pqr 0. Show that R is isomorphic to R/I, × R/Iq.

A: