Definition:Well-Defined/Mapping

Definition
Let $f: S \to T$ be a mapping.

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

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

Let $\phi: S / \RR \to T$ be a mapping such that:


 * $\map \phi {\eqclass x \RR} = \map f x$

Then $\phi: S / \RR \to T$ is well-defined :


 * $\forall \tuple {x, y} \in \RR: \map f x = \map f y$

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

Also see

 * Condition for Mapping from Quotient Set to be Well-Defined