Transitive relation

Definition

Let $R$ be a binary relation on a set $A$. The relation $R$ is transitive when, for all $a, b, c \in A$, $a \mathrel{R} b$ and $b \mathrel{R} c$ imply $a \mathrel{R} c$.

Remarks

The relations $<$, $\le$ and $=$ on $\mathbb{R}$ are transitive, and so is divisibility on $\mathbb{Z}$. "Is a parent of" is not transitive. The smallest transitive relation that contains $R$ is the transitive closure of $R$.

Explanations

L1 human

Transitivity lets you chain steps. From $2 < 5$ and $5 < 9$ you get $2 < 9$. In the graph of a transitive relation, each path of two arrows has a direct arrow as a shortcut.