Composition of Relations is not Commutative

Theorem
Composition of relations is, in general, not commutative:

That is, it is usually the case that:
 * $\RR_1 \circ \RR_2 \ne \RR_2 \circ \RR_1$

for relations $\RR_1$ and $\RR_2$.

Proof
Proof by Counterexample:

Let $\RR_1 := \struct {S, S, R_1}$ and $\RR_2 := \struct {S, S, R_2}$ be relations defined as:

Let:
 * $S = \set {0, 1, 2}$
 * $R_1 = \set {\tuple {0, 1} }$
 * $R_2 = \set {\tuple {1, 2} }$

We have that:
 * $\RR_1 \circ \RR_2 = \struct {S, S, \set {\tuple {0, 2} } }$

while:
 * $\RR_2 \circ \RR_1 = \struct {S, S, \O}$