# 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.

$\blacksquare$

## Sources

applied to a specific instance