Transitive Relation

A relation $R$ over a set $X$ is transitive if for all elements x, y, z in $X$, whenever $R$ relates $x$ to $y$ and $y$ to $z$, then $R$ also relates $x$ to $z$.

$$\begin{equation} \forall x,y,z \in X,if\,xRy\,and\,yRz,\,then\,xRz \end{equation}$$

Tao’s definition:

Given any three objects x, y, and z of the same type, if $x = y$ and $y = z$, then $x = z$.

See also:

• Symmetric Relation

• Reflexive Relation