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 ranges.

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.