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ws803 - Rob Bayer Relations Math 55 Worksheet August 3 2009...

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Rob Bayer Math 55 Worksheet August 3, 2009 Relations 1. Determine whether each of the following relations are symmetric, reflexive, transitive, and/or antisymmetric. (a) < on N (b) on N (c) The relation on R defined by a b iff a - b Z (d) The relation on people given by “is a child of” (e) x = y 2 on R (f) 6 = 2. What’s the probability that a randomly chosen relation on a set of size n is (a) Symmetric? (b) Reflexive? (c) Both? 3. A relation is called irreflexive if ( a, a ) / R for any a (a) Give a natural example of an irreflexive relation (b) Give a (semi-)natural example of a relation that is neither reflexive nor irreflexive 4. Let R 1 be the relation “congruent mod 3” and R 2 be the relation “congruent mod 4,” both on Z . What are the relations (a) R 1 R 2 (b) R 1 R 2 (c) R 2 - R 1 5. True/False. For those that are true, prove it. For those that are false, provide a counterexample. (a) The intersection of two symmetric relations is symmetric (b) The union of two antisymmetric relations is antisymmetric
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