Is {(1, 1), (2, 2)} symmetric and/or antisymmetric?

Question

For relation $$\left\{(1, 1),(2, 2)\right\}$$ decide whether it is symmetric, whether it is antisymmetric, and whether it is transitive?

Answer 1

A relation $R$ can be both symmetric and antisymmetric.

For any such relation, suppose $a R b$. Then by symmetry, $bRa$. By antisymmetry, since $a R b$ and $b R a$, then $ a = b$. Hence, for any such relation, it must be that $ a R b \implies a = b$. You can check that any relation which satisfies $a R b \implies a = b$ is both symmetric and antisymmetric, so that these are equivalent statements. Your relation can easily be checked to satisfy this property (though you haven't stated on what set the relation is, it doesn't matter here), so it is both symmetric and antisymmetric.