Let S ⊆ ℕ and let x ∈ S. We say that x is a minimum element of S if for every y ∈ S, x ≤ y. a) Suppose that S has a minimum element. Prove that this minimum element is unique. b) Prove that if S ⊆ ℕ is nonempty, then S has a minimum element. Hint: For n ∈ ℕ, let P(n) be the statement “If n ∈ S, then S has a minimum element.” Prove that P(n) is true for every n

Q: Exercise 16. Prove the above theorem. Hint: show A ∪ B = A ∪ B − (A ∩ B). Theorem 35…

A: To prove: A∪B=A+B-A∩B

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A: X∪Yc=Xc∩Yc

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Q: 37. Which of these three statements is/are not true? I: Every finite nonempty subset of R has a…

A: We shall answer all the parts and provide justification as to why a particular part is not true. In…

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A:

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Q: True or False. Explain why? Justify your answer mathematically. Use examples, counterexamples,…

A: e)TrueStep 1Let X and Y be finite sets. let ∣X∣=m, ∣Y∣=n f:X→Yis a bijection, that is, f is one one…

Q: d) Suppose X = {1,2, 3} and Y = {5,6}. The number of elements of the set P(X× Y) =

A: Elements in X = 3. Elements in Y = 2 Elements in X × Y = 3 × 2 = 6

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A: Given:

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Q: Prove that if S is a bounded set then S is bounded.

A:

Q: B. Let B and C be the subsets of some universal set X, then prove that (C – B)' = BU C'

A: let B and C be the subsets of some universet x1 then prove that (c-B)' = BUC'

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A: use the basic property of sets .

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A: The universe set contains the elements: p, q, r, s, t, u, v, w, x, y The set A contains the…

Q: A={ y ∣ y ∈ N and 305< y ≤ 335} A has      subsets.  A has              proper subsets.

A: We have, A={ y ∣ y ∈ N and 305&lt; y ≤ 335} Let A and B be two sets. We say that A is a subset of B…

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Q: Let y = x₁ = [7 5 10 and ū= -21 -4 Write y as the sum of two orthogonal vectors, ₁ in Span {u} and ₂…

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Q: Exercise 1. Prove that for sets A, B,C we have C \(AUB) = (C\ A) N (C \ B).

A: We have to show that for sets A, B, C   C ∖(A∪B)=(C∖A)∩(C∖B).

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A: We will construct an example where the sets are not pairwise disjoint but intersection of all sets…

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Q: If D(L AOB) states the set of points located on the OA side containing point B and the OB side…

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Q: Prove the following matrix equation. and show your complete solutions.

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A: Answer :

Q: 22. Let A be a bounded subset of R and let sup A = B. Which of the following is/are not an attribute…

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Q: A set L cR is convex if:

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A:

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A: Part (i) If Then Hence So given statement is wrong . The correct statement is If Then