# Quotient Structure is Well-Defined

## Theorem

Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $\mathcal R$ be a congruence relation on $\left({S, \circ}\right)$.

Let $S / \mathcal R$ be the quotient set of $S$ by $\mathcal R$.

Let $\circ_\mathcal R$ be the operation induced on $S / \mathcal R$ by $\circ$.

Then $\circ_\mathcal R$ is a well-defined operation in the quotient structure $\left({S / \mathcal R, \circ_\mathcal R}\right)$.

## Proof

 $\displaystyle \left[\!\left[{x_1}\right]\!\right]_\mathcal R = \left[\!\left[{x_2}\right]\!\right]_\mathcal R$ $\land$ $\displaystyle \left[\!\left[{y_1}\right]\!\right]_\mathcal R = \left[\!\left[{y_2}\right]\!\right]_\mathcal R$ $\displaystyle \implies \ \$ $\displaystyle x_1 \mathop {\mathcal R} x_2$ $\land$ $\displaystyle y_1 \mathop {\mathcal R} y_2$ Definition of Equivalence Class $\displaystyle \implies \ \$ $\displaystyle \left({x_1 \circ y_1}\right)$ $\mathcal R$ $\displaystyle \left({x_2 \circ y_2}\right)$ Definition of Congruence Relation $\displaystyle \implies \ \$ $\displaystyle \left[\!\left[{x_1 \circ y_1}\right]\!\right]_\mathcal R$ $=$ $\displaystyle \left[\!\left[{x_2 \circ y_2}\right]\!\right]_\mathcal R$ Definition of Equivalence Class

$\blacksquare$