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Homework 10 Solutions W11 - EECS 203 Homework 10 Solutions...

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EECS 203: Homework 10 Solutions Section 8.4 1. (E) 2 When we add the pairs ( x, x ) to the given relation we have all of Z × Z ; in other words, we have the relation that always holds. 2. (E) 6 We form the reflexive closure by taking the given directed graph and appending loops at all vertices at which there are not already loops. a c b d 3. (E) 14 Suppose that the closure C exists. We must show that C is the intersection I of all the relations S that have property P and contain R . Certainly, I C , since C is one of the sets in the intersection. Conversely, by definition of closure, C is a subset of every relation S that has the property P and contains R ; therefore C is contained in their intersection. 4. (M) 22 Since R R * , clearly if 4 ⊂ R , then 4 ⊂ R * . 5. (C) 24 It is certainly possible for R 2 to contain some pairs ( a, a ) . For example , let R = { (1 , 2) , (2 , 1) } 6. (M) 26 (a) We show the various matrices that are invoked. First, A = 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 , A [2] = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0
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