# Why isn’t reflexivity redundant in the definition of equivalence relation?

An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity.

Doesn’t symmetry and transitivity implies reflexivity? Consider the following argument.

For any $a$ and $b$,
$a R b$ implies $b R a$ by symmetry. Using transitivity, we have $a R a$.

Source: Exercise 8.46, P195 of Mathematical Proofs, 2nd (not 3rd) ed. by Chartrand et al

Your argument used the hypothesis that for each $a$, there exists $b$ such that $aRb$ holds. If this is true, then symmetry and transitivity imply reflexivity, but this is not true in general.