Definition:Well-Defined

Definition

Well-Defined Mapping

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 if and only if:

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

Well-Defined Relation

The concept can be generalized to include the general relation $\RR: S \to T$.

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 if and only if:

for arbitrary $x, y, x', y' \in S$ such that:
$x \mathrel \RR x'$
$y \mathrel \RR y'$
we have that:
$\tuple {\eqclass x \RR, \eqclass y \RR} \in \QQ \iff \tuple {\eqclass {x'} \RR, \eqclass {y'} \RR} \in \QQ$

Well-Defined Operation

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 $S / \RR$) if and only if:

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

Also known as

Some sources use the term consistent for well-defined.