# Definition:Diagonal Relation

## Definition

Let $S$ be a set.

The diagonal relation on $S$ is a relation $\Delta_S$ on $S$ such that:

$\Delta_S = \set {\tuple {x, x}: x \in S} \subseteq S \times S$

Alternatively:

$\Delta_S = \set {\tuple {x, y}: x, y \in S: x = y}$

## Also known as

This is sometimes called the equality relation or the identity relation.

It is also referred to it as the diagonal set or diagonal subset (or just the diagonal), but it can be useful to retain the emphasis that it is indeed a relation.

## Also see

Note that the diagonal relation on $S$ is the same as the identity mapping $I_S$ on $S$.

• Results about the diagonal relation can be found here.