# Transitive Relation

##### Updated: October 22, 2020

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: