Definition:Diagonal Relation

Definition

Let $S$ be a set.

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

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

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 $S$.

• Diagonal Relation is Equivalence

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): $\S 1.4.4$