# Definition:Diagonal Relation

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## Contents

## Definition

Let $S$ be a set.

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

- $\Delta_S = \left\{{\left({x, x}\right): x \in S}\right\} \subseteq S \times S$

## Also known as

This is sometimes called the **equality 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 $S$.

## Sources

- Paul R. Halmos:
*Naive Set Theory*(1960)... (previous)... (next): $\S 7$: Relations - Steven A. Gaal:
*Point Set Topology*(1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - Seth Warner:
*Modern Algebra*(1965)... (previous)... (next): $\S 5$ - George McCarty:
*Topology: An Introduction with Application to Topological Groups*(1967)... (previous)... (next): $\text{I}$: Relations - Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(1970)... (previous)... (next): $\text{I}: \ \S 5$: Uniformities - T.S. Blyth:
*Set Theory and Abstract Algebra*(1975)... (previous)... (next): $\S 4$: Example $4.2$ - Steve Awodey:
*Category Theory*(2010)... (previous)... (next): $\S 1.4.4$