# ws804 - Rob Bayer Closures Math 55 Worksheet August 4 2009...

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Rob Bayer Math 55 Worksheet August 4, 2009 Closures 1. Draw the directed graph representing each of the following relations. What are the directed graphs for the symmetric closures of each of these? The transitive closures? (a) { ( a, a ) , ( a, b ) , ( b, c ) , ( c, b ) , ( c, d ) , ( d, a ) , ( d, b ) } (b) { ( a, c ) , ( a, b ) , ( b, c ) , ( c, b ) } 2. Find a simple description of the symmetric closure of each of the following relations: (a) < on N (b) Divisibility relation on Z (c) x = | y | on R 3. Find the transitive closure of each of the following relations. Your description should be reasonably simple: (a) “is a parent of” on the set of all people (b) | x - y | = 1 on R (c) “there is a flight from x to y ” on the set of all cities with airports 4. Does it make sense to talk about the antisymmetric closure of a relation? What about the irreflexive closure? 5. True/False. For those that are true, prove it. Throughout, t ( R ) will denote the transitive closure of R . (a) If R is reflexive, then t ( R ) is reflexive too.
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