Quotient Mapping on Structure is Epimorphism

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

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


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

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

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

Next we show that this is a homomorphism:

Thus the morphism property is shown to hold.

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