Diagonal Relation is Equivalence

Theorem

The diagonal relation $\Delta_S$ on a set $S$ is always an equivalence in $S$.

Proof

Checking in turn each of the criteria for equivalence:

Reflexive

 $\, \displaystyle \forall x \in S: \,$ $\displaystyle \tuple {x, x}$ $\in$ $\displaystyle \Delta_S$ Definition of Diagonal Relation

So $\Delta_S$ is reflexive.

$\Box$

Symmetric

 $\, \displaystyle \forall x, y \in S: \,$ $\displaystyle \tuple {x, y}$ $\in$ $\displaystyle \Delta_S$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle y$ Definition of Diagonal Relation $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle x$ Equality is Symmetric $\displaystyle \leadsto \ \$ $\displaystyle \tuple {y, x}$ $\in$ $\displaystyle \Delta_S$ Definition of Diagonal Relation

So $\Delta_S$ is symmetric.

$\Box$

Transitive

 $\, \displaystyle \forall x, y, z \in S: \,$ $\displaystyle \tuple {x, y}$ $\in$ $\displaystyle \Delta_S \land \tuple {y, z} \in \Delta_S$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle y \land y = z$ Definition of Diagonal Relation $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle z$ Equality is Transitive $\displaystyle \leadsto \ \$ $\displaystyle \tuple {x, z}$ $\in$ $\displaystyle \Delta_S$ Definition of Diagonal Relation

So $\Delta_S$ is transitive.

$\blacksquare$