# transitive relation

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 $1\neq 1$.

Title transitive relation TransitiveRelation 2013-03-22 12:15:52 2013-03-22 12:15:52 yark (2760) yark (2760) 14 yark (2760) Definition msc 03E20 Reflexive Symmetric Antisymmetric transitivity transitive