Unformatted text preview: C ) implies x ∈ ( A ∪ B ) ∩ ( A ∪ C ) . To prove part 2, we must prove that x ∈ ( A ∪ B ) ∩ ( A ∪ C ) implies x ∈ A ∪ ( B ∩ C ) . In this case the proof is simple, since it just follows the above proof in reverse. The details are left as an exercise. ± The above proof is an example of the generality often required to prove a property about sets: it uses arbitrary sets and arbitrary elements of such a set. In contrast, to show that a property is false, it is enough to Fnd one counterexample. Such counterexamples should be as simple as possible, to illustrate that a statement is not true with minimum effort to the reader. P ROPOSITION 2.7 The following statements are not true: 1. A ∪ ( B ∩ C ) = ( A ∩ B ) ∪ C ; 2. A ∪ ( B ∩ C ) = ( A ∩ B ) ∪ ( A ∩ C ) . 9...
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 Spring '09
 Koskesh
 Math, Logic, Sets, arbitrary sets

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