Diagonal Relation is Universally Compatible

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Theorem

The diagonal relation $\Delta_S$ on a set $S$ is universally compatible with every operation on $S$.


Proof

Let $\left({S, \circ}\right)$ be any algebraic structure.

\(\displaystyle \) \(\) \(\displaystyle x_1 \Delta_S x_2 \land y_1 \Delta_S y_2\)
\(\displaystyle \) \(\implies\) \(\displaystyle x_1 = x_2 \land y_1 = y_2\) Definition of Diagonal Relation
\(\displaystyle \) \(\implies\) \(\displaystyle x_1 \circ y_1 = x_2 \circ y_2\) (consequence of equality)
\(\displaystyle \) \(\implies\) \(\displaystyle \left({x_1 \circ y_1}\right) \Delta_S \left({x_2 \circ y_2}\right)\) Definition of Diagonal Relation

$\blacksquare$