Question 3 (10 Points): Set theory and logics (02) - a) Identify the truth value of...

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Question 3 (10 Points): Set theory and logics (02) - a) Identify the truth value of the proposition "For arbitrary sets A, B, C, D, (A B) (C D) = (A x C) (B x D)". Prove your claim. (5 Points) b) For arbitrary sets A, B, C, prove that (A B) - C = A - (BUC). (5 points) Question 4 (15 Points): Relations (01, 02) a) For each of the following relations, prove or disprove that it is 1) reflexive, 2) symmetric, 3) transitive. (10 Points) i) For a, b R, the relation R defined as (a, b) R if and only if 3 n Z, n 1 so that a = b n. ii) For a, b c N, the relation R defined as (a,b) R if and only if anb {1,2,3,4}. b) Let R, R' be relations on the same set A. Prove that, if R, R' are both transitive, then RnR' is also transitive. (5 Points) 2 Question 3 (10 Points): Set theory and logics (02) - a) Identify the truth value of the proposition "For arbitrary sets A, B, C, D, (A B) (C D) = (A x C) (B x D)". Prove your claim. (5 Points) b) For arbitrary sets A, B, C, prove that (A B) - C = A - (BUC). (5 points) Question 4 (15 Points): Relations (01, 02) a) For each of the following relations, prove or disprove that it is 1) reflexive, 2) symmetric, 3) transitive. (10 Points) i) For a, b R, the relation R defined as (a, b) R if and only if 3 n Z, n 1 so that a = b n. ii) For a, b c N, the relation R defined as (a,b) R if and only if anb {1,2,3,4}. b) Let R, R' be relations on the same set A. Prove that, if R, R' are both transitive, then RnR' is also transitive. (5 Points) 2

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a The proposition For arbitrary sets A B C D A B x C D A x C B x D Prove your claim 5 Points The proposition is false To prove this lets consider a counterexample Let A 1 2 B 2 3 C 2 3 and D 3 4 Then ... View the full answer