Prove the following A relation R is asymmetric if and only i

Prove the following:

A relation R is asymmetric if and only if R is irreflexive and antisymetric.

Hint: write the definition of what it means to be asymmetric, then \"add\" a contradiction.

Prove the following: A relation R is asymmetric if and only if R is irreflexive and antisymetric. Note: a relation R on the set A is irreflexive if for every That is, R is irreflexive if no element in A is related to itself. Hint: write the definition of what it means to be asymmetric, then add a contradiction.

Solution

An asymmetric relation must be antisymmetric, since the hypothesis of the condition for antisymmetry is false if the relation is asymmetric. The relation {(a,a)} on {a} is antisymmetric but not asymmetric, however, so the answer to the second question is no. In fact, it is easy to see that R is asymmetric if and only if R is antisymmetric and irreflexive.