Assignment 3 Solutions - Set Theory (MAT2000) - Math 2000 Set Theory Assignment 3 Due Date Oct 1 2012 Problem 1 Let A B C be sets Determine if the

# Assignment 3 Solutions - Set Theory (MAT2000) - Math 2000...

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Math 2000: Set Theory Assignment 3 Due Date: Oct. 1 2012 Problem 1. Let A, B, C be sets. Determine if the following assertions are true or false. Provide a proof or counterexample in each case. a) ( A B ) ( B C ) A B b) ( A B ) ( B C ) A B Proof. a) False. Take as counterexample A = { a, 1 , 2 , 3 } , B = { b, 1 , 2 , 3 , 4 , 5 } , C = { c, 1 , 2 , 3 , 4 , 5 } . Work through why this fails. Of course there are other counterexamples , . b) False. Take as counterexample A = { 1 , 2 , 3 } , B = { a, b, c } , C = { 0 , 1 , 2 , 3 , 4 } . Work through why this fails. Of course there are other counterexamples , . Problem 2. Let A, B, C be subsets of U . Prove A ( B C ) = ( A B ) ( A C ) . Proof. x A ( B C ) ( x A ) ( x B C ) ( x A ) ( ( x B ) ( x C ) ) ( ( x A ) ( x B ) ) ( ( x A ) ( x C ) ) ( ( x A B ) ( ( x A C ) ) x ( A B ) ( A C ) . (1) Problem 3. Let A , B , and C be sets in some universe U . Prove that A ( B Δ C ) = ( A B )Δ( A C ) by showing that A ( B Δ C ) and ( A B )Δ( A C ) are subsets of each other. Proof. Suppose x A ( B Δ C ). Then x A and x B Δ C , so x A , x ( B C ), and x / B C . So if x B then x / C , in which case x A B and x / A C , so x ( A B ) ( A C