Equivalence Relation/Examples/Non-Equivalence/Greater Than

Example of Relation which is not Equivalence
Let $\R$ denote the set of real number.

Let $>$ denote the usual relation on $\R$ defined as:
 * $\forall \tuple {x, y} \in \R \times \R: x > y \iff \text { $x$ is (strictly) greater than $y$}$

Then $>$ is not an equivalence relation.

Proof
We have that $>$ is transitive:
 * $x > y, y > z \implies x > z$

But $>$ is not reflexive:
 * $\forall x: x \not > x$

$>$ is not symmetric:
 * $x > y \implies y \not > x$

So $\sim$ is not symmetric.

So $\sim$ is not an equivalence relation.