Let R be a relation on a set A Prove or disprove If R is an

Let R be a relation on a set A. Prove or disprove : If R is antisymmetric, then R is irreflexive.

Solution

R is both antisymmetric and reflexive, as desired.......

Find a counterexample with the set A = {1, 2}.

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We disprove the conjecture by counterexample. That is, we construct a relation R on the given set A that is antisymmetric (satisfying the hypothesis), but NOT irreflexive (contradicting the conclusion). That is, we will construct R so that R is both antisymmetric and reflexive.

Consider R = {(1, 1), (2, 2)}.

Certainly, R is reflexive, since xRx for each x in A.
Furthermore, R is antisymmetric, since if xRy and yRx, then x = y.

Thus, R is both antisymmetric and reflexive, as desired.