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The relation < on the integers is also antisymmetric; since a < b and b < a is impossible, the antisymmetry condition is vacuously true.
Note that antisymmetry is not the opposite of symmetry (aRb implies bRa). There are relations which are both symmetric and antisymmetric (equality), there are relations which are neither symmetric nor antisymmetric (divisibility on the integers), there are relations which are symmetric and not antisymmetric (congruence modulo n), and there are relations which are not symmetric but antisymmetric (lessthan on the integers).
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