# Diagonal Relation is Ordering and Equivalence

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## Theorem

Let $\struct {S, \Delta_S}$ be a relational structure where $\Delta_S$ is the diagonal relation, defined as:

- $\forall x, y \in S: \paren {x, y} \in \Delta_S \iff x = y$

Then $\Delta_S$ is the **only** relation on $S$ which is both an equivalence and an ordering.

## Proof

From Trivial Ordering is Universally Compatible and Diagonal Relation is Equivalence we know that the diagonal relation possesses these properties.

We now need to show that it is the **only** relation on $S$ which possesses these properties.

Both an equivalence and an ordering are reflexive and transitive.

Also:

- An equivalence relation is symmetric
- An ordering is antisymmetric.

The result follows from Relation is Symmetric and Antisymmetric iff Coreflexive.

$\blacksquare$

## Sources

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$: Exercise $1$