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 $S$.

• Results about the diagonal relation can be found here.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 5$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): $\text{I}$: Relations
• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{I}: \ \S 5$: Uniformities
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$: Example $4.2$
• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.4$