Test 2 Solutions - W07/CS 191 Suggested Solutions to TEST...

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Unformatted text preview: W07/CS 191 : Suggested Solutions to TEST TWO Discrete Structures I 03/22/2007 OPEN BOOK Name_____________________ There are 5 problems with 20 points each. 1 Let X = { a , b , c , d }. (a) List all members of its power set P . Answer: P = { ∅ , { a }, { b }, { c }, { d }, { a , b }, { a , c }, { a , d }, { b , c }, { b , d }, { c , d }, { a , b , c }, { a , b , d }, { a , c , d }, { b , c , d }, { a , b , c , d }} (b) If we define a relation R on the power set P such that A R B if | A | = | B |, where A , B ∈ P . For example, { a , b } R { c , d } because |{ a , b }| = |{ c , d }| = 2. Prove that R is an equivalent relation. Answer: Because for and set A ∈ P , | A | = | A |, so, R is reflexive. If A R B , then we have | A | = | B | which implies | B | = | A | and hence B R A . So, R is symmetric. If A R B and B R B , then we have | A | = | B | and | B | = | C |, which implies | A | = | C | and A R C . Therefore R is transitive....
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Test 2 Solutions - W07/CS 191 Suggested Solutions to TEST...

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