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14+PS3

# 14+PS3 - 1 Handout#14 CS103 Robert Plummer Problem Set#3Due...

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1 Handout #14 CS103 April 11, 2011 Robert Plummer Problem Set #3—Due Monday, 4/18 in class Some of the following problems involve proofs about sets. Here is a sample of the type of proof you should use. (Do not use Venn diagrams in your proof.) Suppose A and B are sets. Prove that A (A B) = A. PROOF: We first show that A (A B) A. If x A (A B), then x A and x (A B) by the definition of intersection. By simplification, x A. So A (A B) A by definition of subset. We next show that A A (A B). If x A, then x (A B) by definition of union, and x A (A B) by definition of intersection. So A A (A B). Two sets are equal if each is a subset of the other, so A (A B) = A. 1. Suppose A and B are sets. Prove that (A B) - A = B - (A B). 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. 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.

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14+PS3 - 1 Handout#14 CS103 Robert Plummer Problem Set#3Due...

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