# Diagonal Relation is Symmetric

The diagonal relation $\Delta_S$ on a set $S$ is a symmetric relation in $S$.
 $\, \displaystyle \forall x, y \in S: \,$ $\displaystyle \tuple {x, y}$ $\in$ $\displaystyle \Delta_S$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle y$ $\quad$ Definition of Diagonal Relation $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle x$ $\quad$ Equality is Symmetric $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle \tuple {y, x}$ $\in$ $\displaystyle \Delta_S$ $\quad$ Definition of Diagonal Relation $\quad$
So $\Delta_S$ is symmetric.
$\blacksquare$