# Equality of Relations

## Theorem

Two relations $\mathcal R_1 \subseteq S_1 \times T_1, \mathcal R_2 \subseteq S_2 \times T_2$ are equal iff:

• $S_1 = S_2$
• $T_1 = T_2$
• $\left({s, t}\right) \in \mathcal R_1 \iff \left({s, t}\right) \in \mathcal R_2$.

It is worth labouring the point that for two relations to be equal, not only must their domains be equal, but so must their codomains.

Some sources refer to this concept between two relations as being equivalence, rather than equality.

## Proof

This follows from set equality and Equality of Ordered Pairs.

$\blacksquare$