Definition:Diagonal Relation
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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
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): Chapter $\text{I}$: Sets and Functions: Relations
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings: Example $4.2$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Uniformities
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.4$