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 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| ? n2

Explanation / Answer

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

b) FALSE

R2 is a relation on Z+ where aRb if a|b and R1 is a relation on Z+ where aRb if a is a proper divisor of b

c) FALSE

R2 is a relation on Z+ where aRb if a|b and R1 is a relation on Z+ where aRb if a is a proper divisor of

d) TRUE

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