5. For any set X, let P(X) denote the power set of X. Prove or disprove the following statements (In other words, determine whether each of the following statements is true or false, and prove your claim. If the statement is true, give a proof. If the statement is false, prove that the negation is true, i.e. give a counterexample). (a) For any sets A and B, P(AN B) = P(A) N P(B). (b) For any sets A and B, P(AU B) = P(A) U P(B). (c) For any sets A, B, and C, A\ (BUC) = (A\ B)U (A \ C). (d) For any sets A, B, C, if An BC C, then (A \C)NB=0.

5. For any set X, let P(X) denote the power set of X. Prove or disprove the following statements (In other words, determine whether each of the following statements is true or false, and prove your claim. If the statement is true, give a proof. If the statement is false, prove that the negation is true, i.e. give a counterexample). (a) For any sets A and B, P(AN B) = P(A) N P(B). (b) For any sets A and B, P(AU B) = P(A) U P(B). (c) For any sets A, B, and C, A\ (BUC) = (A\ B)U (A \ C). (d) For any sets A, B, C, if An BC C, then (A \C)NB=0.

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

ChapterP: Prerequisites

SectionP.2: Real Numbers

Problem 3E: Express the set of real numbers between but not including 2 and 7 as follows. (a) In set-builder...

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Question

the last one . Please write neatly and logiclaly

5. For any set \(X\), let \(\mathcal{P}(X)\) denote the power set of \(X\). Prove or disprove the following statements (In other words, determine whether each of the following statements is true or false, and prove your claim. If the statement is true, give a proof. If the statement is false, prove that the negation is true, i.e. give a counterexample).

(a) For any sets \(A\) and \(B\), \(\mathcal{P}(A \cap B) = \mathcal{P}(A) \cap \mathcal{P}(B)\).

(b) For any sets \(A\) and \(B\), \(\mathcal{P}(A \cup B) = \mathcal{P}(A) \cup \mathcal{P}(B)\).

(c) For any sets \(A\), \(B\), and \(C\), \(A \setminus (B \cup C) = (A \setminus B) \cup (A \setminus C)\).

(d) For any sets \(A\), \(B\), \(C\), if \(A \cap B \subseteq C\), then \((A \setminus C) \cap B = \emptyset\).

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