Diagonal Relation is Universally Compatible

Theorem

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

Proof

The validity of the material on this page is questionable. In particular: It needs to be pointed out that the definition of universally compatible relation may not be correct as such. I will address it when I work up to that chapter in my book. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues. 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 {{Questionable}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Let $\struct {S, \circ}$ be any algebraic structure.

\(\ds \)

\(\)

\(\ds x_1 \Delta_S x_2 \land y_1 \Delta_S y_2\)

\(\ds \)

\(\leadsto\)

\(\ds x_1 = x_2 \land y_1 = y_2\)

Definition of Diagonal Relation

\(\ds \)

\(\leadsto\)

\(\ds x_1 \circ y_1 = x_2 \circ y_2\)

(consequence of equality)

\(\ds \)

\(\leadsto\)

\(\ds \paren {x_1 \circ y_1} \mathrel {\Delta_S} \paren {x_2 \circ y_2}\)

Definition of Diagonal Relation

$\blacksquare$