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

 * Definition:Equals
 * Diagonal Relation is Equivalence

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