Transitive relation

Transitive relation

Type	Binary relation
Field	Elementary algebra
Statement	A relation ${\displaystyle R}$ on a set ${\displaystyle X}$ is transitive if, for all elements ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$ in ${\displaystyle X}$, whenever ${\displaystyle R}$ relates ${\displaystyle a}$ to ${\displaystyle b}$ and ${\displaystyle b}$ to ${\displaystyle c}$, then ${\displaystyle R}$ also relates ${\displaystyle a}$ to ${\displaystyle c}$.
Symbolic statement	${\displaystyle \forall a,b,c\in X:(aRb\wedge bRc)\Rightarrow aRc}$

In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.

Every partial order and every equivalence relation is transitive. For example, less than and equality among real numbers are both transitive: If a < b and b < c then a < c; and if x = y and y = z then x = z.