Equality of Relations

Theorem
Let $\mathcal R_1$ and $\mathcal R_2$ be relations on $S_1 \times T_1$ and $S_2 \times T_2$ respectively.

Then $\mathcal R_1$ and $\mathcal R_2$ are equal :


 * $S_1 = S_2$
 * $T_1 = T_2$
 * $\tuple {s, t} \in \mathcal R_1 \iff \tuple {s, t} \in \mathcal R_2$

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.

Proof
This follows from set equality and Equality of Ordered Pairs.

A relation on $S \times T$ is an ordered triple, thus:

From Equality of Ordered Tuples, $\mathcal R_1 = \mathcal R_1$ :

Then we have that $R_1$ and $R_2$ are sets of ordered pairs.

Thus by definition of set equality:
 * $R_1 = R_2 \iff \paren {\forall \tuple {x, y} \in \S_1 \times T_2: \tuple {x, y} \in R_1 \iff \tuple {x, y} \in R_2}$

Hence the result.

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