Doesn't symmetry and transitivity imply reflexitivity?

Question

I've been wondering recently the following:

Let $\sim $ be a symmetric and transitive relation defined on $S $. Let $a \sim b $, which implies $b \sim a $ by symmetry, and by transitivity, $a \sim a $. Hence, $\sim $ is reflexive.

I can't think of any counterexamples, although any are welcome. Also, if there is some counterexample, where could be the flaw in my reasoning be?

Answer 1

You're assuming that there exists a $b$ such that $a\sim b$ in your argument. If you don't know the relation is reflexive, that $b$ may not exist. Put differently, reflexivity in the presence of symmetry and transitivity is equivalent to each element being equivalent to some other one

Answer 2

Reflexivity is a~a. You are using symmetry as reflexivity. As vadim123, noted, symmetry and transitivity do not imply reflexivity. The empty relation is a counterexample.