Exercise 10. Show that if AC B, then ACB.

Q: show that }I f,lm) Ck-m) 2.

A: For, ∑m=0kf1mf2k-m, substitute k-m=n or m=k-n: ∑m=0kf1mf2k-m=∑k-n=0kf1k-nf2k-k-n…

Q: Let H≤G & KG. Show that HnKH. Let H◄ G & K G and assume that Hn K = {e}. Show that if aeH & beK then…

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Q: Let a, b, x, y ∈Zbe such that ax+by= 11. May we conclude that gcd(a, b) = 11?Why or why not? (Write…

A: Consider the given information:Let a,b,x,y∈ℤ be such that ax+ by=11.

Q: 2 Let A | and U = H. Then A2017U is equal to = ... A) B

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Q: n ∈ Z, d ∈ N} and define (a, b) ≡ (c, d) to mean ad = bc 1. For any (n,d) we have (n,d) ≡ (n,d) 2.…

A: Equivalence relation:  Equivalence relations are binary relations that are reflexive, symmetric, and…

Q: Is the following true or false? Z∩N=Z

A: Z is the set of all integers, i.e.  Z={ ..., -3, -2, -1, 0, 1, 2, 3, ...} And N is the set of all…

Q: 1. Let P(A) = 0.3 and P(B) = 0.6. a) Find P(AUB) when A and B are independent. b) Find P(A/B) when A…

A: P(A) = 0.3P(B) = 0.6

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Q: 2.If alb and alc then al Cbx tcy) for all X>YEZ.

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Q: 8. Let dgcd(135, 351). Find integers m and n such that 135m + 351n = d.

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Q: Let A,BE Mmna Shaw that vTAw= V' Bw for all VER™ end WEIR" if and only if LA=B. %3D

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Q: Ema thinks that (u) xw, exists. Paul disagrees with her. Who is right? Explain why.

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Q: 1,2, ..., let A; = {i,i + 1, i + 2, …. }. Find ỦA. For i i) ii) N A, i=1 i=1

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Q: Suppose that a divides b and that c divides b. Suppose that gcd(a, c) = 1. Prove that ac divides b.

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Q: Suppose E = {a, b, c, d, e, f, g, h, i, j}, %3D A= {a, b, c, d, e, f, g}, B= {b, d, f, i, j}, C =…

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Q: 9f a,b,c,d EN & a = b frod n) C=d(modn) sherw that bd(radn)

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Q: 14. Prove that AC B if and only if An BC= 0.

A: The objective is to prove the above question.

Q: Let A, B, C subset U. Suppose that n(U)= 90, n(A)= 40, n(B)= 55, and n(C)= 20. Which of the…

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Q: let d= gedlab) then prone d C and C=ab

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Q: Let A = {c, d, f, g), B = {f, j}, and C = {d, g). BcA A True B) False

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Q: Suppose a, b e (0, 5) and a b – a.

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Q: Let S := [a, b] with a < b. Then Acc(S) = [a,b]. Ja, bl. □R□ none of the previous.

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Q: Exercise 3. Suppose that A and B are independent. Argue that A' and B' are also independent.

A: P(A⋂B)=P(A)P(B)  when A  and B are independent.

Q: Let u (41, U2, Uz). Explain why u•u > 0. When is u•u 0?

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Q: if a2+aEccR; Fos all fiso:e that aER, Ris camutative. then

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Q: 4 Show that DE B0) n s+ Ø

A: We will use basic knowledge of metric space (or general topology) to answer this question.…

Q: Compute the SVD of QA = 3 2 3 322

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Q: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

A: Given that gcd(a,b)=1 If possible , let us assume that  gcd(a+b,ab)=d,d≠1 Then the number d must…

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Q: Suppose gcd(ab,c)=1. Prove that abc | d if, and only if ab | d and c | d.

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Q: a) In1/e b) eIn5 c) In1/3 d) 2Ine e) 5In1 and F) Ine1/3

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