Quotient Mapping on Structure is Epimorphism

Theorem
Let $\RR$ be a congruence relation on an algebraic structure $\struct {S, \circ}$.

Then the quotient mapping from $\struct {S, \circ}$ to the quotient structure $\struct {S / \RR, \circ_\RR}$ is an epimorphism:


 * $q_\RR: \struct {S, \circ} \to \struct {S / \RR, \circ_\RR}: \forall x, y \in S: \map {q_\RR} {x \circ y} = \map {q_\RR} x \circ_\RR \map {q_\RR} y$

This is sometimes called the canonical epimorphism from $\struct {S, \circ}$ to $\struct {S / \RR, \circ_\RR}$.

Proof
The quotient mapping $q_\RR: S \to S / \RR$ is the canonical surjection from $S$ to $S / \RR$.

Next we show that this is a homomorphism:

Thus the morphism property is shown to hold.

So the quotient mapping $q_\RR: \struct {S, \circ} \to \struct {S / \RR, \circ_\RR}$ has been shown to be a homomorphism which is a surjection, and is thus an epimorphism.