Diagonal Relation is Right Identity

Theorem

Let $\RR \subseteq S \times T$ be a relation on $S \times T$.

Then:

$\RR \circ \Delta_S = \RR$

where $\Delta_S$ is the diagonal relation on $S$, and $\circ$ signifies composition of relations.

Proof

This has to be rewritten.
In particular: In light of the more relaxed definition of composite of relation
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by doing so.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Rewrite}} from the code.

We use the definition of relation equality, as follows:

Equality of Codomains

The codomains of $\RR$ and $\RR \circ \Delta_S$ are both equal to $T$ from Codomain of Composite Relation.

Equality of Domains

The domains of $\RR$ and $\RR \circ \Delta_S$ are also easily shown to be equal.

From Domain of Composite Relation:

$\Dom {\RR \circ \Delta_S} = \Dom {\Delta_S}$

But from the definition of the diagonal relation:

$\Dom {\Delta_S} = \Img {\Delta_S} = S$

Equality of Relations

The composite of $\Delta_S$ and $\RR$ is defined as:

$\RR \circ \Delta_S = \set {\tuple {x, z} \in S \times T: \exists y \in S: \tuple {x, y} \in \Delta_S \land \tuple {y, z} \in \RR}$

But by definition of the diagonal relation on $S$, we have that:

$\tuple {x, y} \in \Delta_S \implies x = y$

Hence:

$\RR \circ \Delta_S = \set {\tuple {y, z} \in S \times T: \exists y \in S: \tuple {y, y} \in \Delta_S \land \tuple {y, z} \in \RR}$

But as $\forall y \in S: \tuple {y, y} \in \Delta_S$, this means:

$\RR \circ \Delta_S = \set {\tuple {y, z} \in S \times T: \tuple {y, z} \in \RR}$

That is:

$\RR \circ \Delta_S = \RR$

Hence the result.

$\blacksquare$

Also see

Diagonal Relation is Left Identity

Sources

1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 5$: Composites and Inverses of Functions: Exercise $5.8 \ \text{(d)}$