# 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:

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

### Class Theory

In the context of class theory, the definition follows the same lines:

Let $V$ be a basic universe.

The diagonal relation on $V$ is the relation $\Delta_V$ on $V$ defined as:

$\Delta_V = \set {\tuple {x, x}: x \in V}$

Alternatively:

$\Delta_V = \set {\tuple {x, y}: x, y \in V: x = y}$

## Also known as

The diagonal relation can also be referred to as the equality relation or the identity relation.

It is also referred to it as:

the diagonal set or class
the diagonal subset or subclass.

Some sources call it just the diagonal.

However, $\mathsf{Pr} \infty \mathsf{fWiki}$'s position is that it can be useful to retain the emphasis that it is indeed a relation.

## Also see

• Results about the diagonal relation can be found here.