For each of the following statements about relations on a set A, where |A| = n,

Question

For each of the following statements about relations on a set A, where |A| = n, determine whether the statement is true or false. If it is false, give a counterexample. a) If R is a relation on A and | R | ? n, then R is reflexive. b) If R 1, R 2 are relations on A and R 2 ?R 1, then R 1 reflexive (symmetric, antisymmetric, transitive) ?_2 reflexive (symmetric, antisymmetric, transitive). c) If R 1, R 2 are relations on A and R 2 ?R 1, then R 2 reflexive (symmetric, antisymmetric, transitive) ?R 1 reflexive (symmetric, antisymmetric, transitive). d) If R is an equivalence relation on A, then n ? | R | ? n2.

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