A relation $\mathcal{R}$ on a set $A$ is transitive if and only if $\forall x,y,z \in A$ $(x\mathcal{R}y \land y\mathcal{R}z) \rightarrow (x\mathcal{R}z)$

For example, the ``is a subset of'' relation $\subseteq$ on any set of sets is transitive. The ``less than'' relation $<$ on the set of real numbers is also transitive.

The ``is not equal to'' relation $\neq$ on the set of integers is not transitive, because $1\neq 2$ and $2\neq 1$ does not imply $1\neq 1$