Definition:Well-Defined/Operation

Definition
Let $\struct {S, \circ}$ be an algebraic structure.

Let $\RR$ be a congruence for $\circ$.

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

Let $\struct {S / \RR, \circ_\RR}$ be the quotient structure defined by $\RR$, where $\circ_\RR$ is defined as:


 * $\eqclass x \RR \circ_\RR \eqclass y \RR = \eqclass {x \circ y} \RR$

Then $\circ_\RR$ is well-defined (on $S / \RR$) :


 * $x, x' \in \eqclass x \RR, y, y' \in \eqclass y \RR \implies x \circ y = x' \circ y'$