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