What is wrong with the following argument?

Question

Say we have a set $X$ and an equivalence relation $C$ on $X$. Why do we need reflexivity?

Let $x,y \in X$ with $xCy $. By symmetry we obtain $yCx$.

Applying now transitivity, we have $xCx$.

So, we have reflexivity from symmetry and transitivity.

Answer 1

You first need to assume there exists, $xCy$. Indeed it's not necessarily true. Think about an extreme cases that $X\neq \emptyset$ and define a relation $C$ on $X$ such that $C$ contains nothing, that is, no element of $X$ is in the relation, then clearly the relation is symmetry and transitive, but not reflexive.