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4. Suppose that A, B, C are subset of a set U, and recall the definitions of the elementary set operations n, U, ⇒, \, and ()* from Chapter 11, as well as their characterizations. Prove the following statements, following any method that you prefer. (a) A⇒ B=A* UB. (b) (An B)* = A*UB*. (c) If A ⇒ 0=U \ A = A*. (d) A⇒ (B⇒ C) = (An B) ⇒ C.

4. Suppose that A, B, C are subset of a set U, and recall the definitions of the elementary set operations n, U, ⇒, \, and ()* from Chapter 11, as well as their characterizations. Prove the following statements, following any method that you prefer. (a) A⇒ B=A* UB. (b) (An B)* = A*UB*. (c) If A ⇒ 0=U \ A = A*. (d) A⇒ (B⇒ C) = (An B) ⇒ C.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. Suppose that \( A, B, C \) are subsets of a set \( U \), and recall the definitions of the elementary set operations \( \cap, \cup, \Rightarrow, \backslash, \) and \(( )^*\) from Chapter 11, as well as their characterizations. Prove the following statements, following any method that you prefer. (a) \( A \Rightarrow B = A^* \cup B \). (b) \((A \cap B)^* = A^* \cup B^*\). (c) If \( A \Rightarrow \emptyset = U \), then \( U \backslash A = A^* \). (d) \( A \Rightarrow (B \Rightarrow C) = (A \cap B) \Rightarrow C \).
Transcribed Image Text:4. Suppose that \( A, B, C \) are subsets of a set \( U \), and recall the definitions of the elementary set operations \( \cap, \cup, \Rightarrow, \backslash, \) and \(( )^*\) from Chapter 11, as well as their characterizations. Prove the following statements, following any method that you prefer. (a) \( A \Rightarrow B = A^* \cup B \). (b) \((A \cap B)^* = A^* \cup B^*\). (c) If \( A \Rightarrow \emptyset = U \), then \( U \backslash A = A^* \). (d) \( A \Rightarrow (B \Rightarrow C) = (A \cap B) \Rightarrow C \).
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