A binary relation $\leq$ on a set $X$ is a preorder if:
(Reflexivity) For all $x \in X$,
\[x \leq x.\](Transitivity) For all $x$, $y$, $z \in X$,
\[x \leq y \,\land\, y \leq z \enspace\rimp\enspace x \leq z.\]
If the symbol $\leq$ denotes a preorder, then $<$ refers to the relation defined by
\[x < y \enspace\lrimp\enspace x \leq y \,\land\, x \neq y.\]