Definition:Well-Defined/Relation

Definition
Let $\RR$ be an equivalence relation on $S$.

For $x \in S$, let $\eqclass x \RR$ denote the equivalence class of $x$ under $\RR$.

Let $S / \RR$ be the quotient set of $S$ determined by $\RR$.

Let $\QQ$ be a relation on $S / \RR$.

Then $\QQ$ is well-defined :


 * for arbitrary $x, y, x', y' \in S$ such that:


 * $x \mathrel \RR x'$
 * $y \mathrel \RR y'$


 * we have that:
 * $\tuple {\eqclass x \RR, \eqclass y \RR} \in \QQ \iff \tuple {\eqclass {x'} \RR, \eqclass {y'} \RR} \in \QQ$

Also known as
Some sources use the term consistent for well-defined.