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Unformatted text preview: 1 Handout #22 CS103 April 18, 2011 Robert Plummer Problem Set #3 Solutions 1. Suppose A and B are sets. Prove that (A B)  A = B  (A B). PROOF: We will prove equality by showing that LHS RHS and RHS LHS. [We'll put some whitespace in the first half of the proof to show how it improves readability.] Suppose x ((A B) – A). Then (x A x B) x A by the definition of union and set difference. Since x A and x A cannot both be true, we have x B x A. If x A, then x A B, so we have x B x (A B), which means x (B – (A B)) by the definition of set difference. So LHS RHS. For the second half of the proof, suppose x (B – (A B)). Then x B x (A B) by the definition of set difference. Thus x B (x A x B) by definition of intersection, and x B x A since x B and x B cannot both be true. Since x B, x A B, so we have x (A B) x A and x ((A B) – A) ) by the definition of set difference. So RHS LHS. ■ 2. Consider the following three conditions: (i) A B (ii) A C (iii) A B  C = Can there exist sets A, B, and C that satisfy all three conditions? If so, provide an example. If not, provide a proof to that effect. Example: A = {1, 2}, B = {2, 3}, C = {2, 4}. Or, any nonempty sets where A = B = C. 3. If A and B are sets, is it possible that A B and A B? Give an example or prove that this is not possible. Example: A = {1, 2}, B = {1, 2, {1, 2}} 4. Suppose that A and B are sets. Prove that B A = A if and only if A B. [Indenting can improve readability, and placing a formula on a separate line makes it convenient to give it a number for later reference. Lower case Roman numerals are often used for this purpose.] PROOF. First the forward direction ( ): Suppose that (B A) = A (i) and suppose that x A. Then by (i), x B A., so x B. T h u s A B. Now the backward direction ( ): Suppose that A B (ii) and suppose that x B A. Then x A. Now suppose x A. Then x B by (ii), and x B A. T h u s B A = A. ■ 2 5. Suppose A and B are sets and that A B. Suppose C is a set such that A C = B C. Prove that C = . ["Wlog" stands for "without loss of generality." Below, it does not matter whether we suppose x A and x B, or x A and x B. So, we just pick one and use "wlog" to mean that the proof would be the same either way....
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This note was uploaded on 06/01/2011 for the course EE 103 taught by Professor Plummer during the Spring '11 term at Stanford.
 Spring '11
 PLUMMER

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