Quotient Structure is Well-Defined

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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

$\blacksquare$


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