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 <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /?>

a) If R is a relation on A and |R| ? n, then R is reflexive.

b) If R1, R2 are relations on A and R2 ? R1, then R1 reflexive (symmetric, antisymmetric, transitive) ?R2 reflexive (symmetric, antisymmetric, transitive).

c) If R1, R2 are relations on A and R2 ? R1, then R2 reflexive (symmetric, antisymmetric, transitive) ? R1 reflexive (symmetric, antisymmetric, transitive).

d) If R is an equivalence relation on A, then n ? |R| ? n^2

Explanation / Answer

a) True b) False c) True d) True

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