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 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:

$\tuple {\eqclass x \RR, \eqclass y \RR} \in \QQ \iff \tuple {\eqclass {x'} \RR, \eqclass {y'} \RR} \in \QQ$

Also known as

Some sources use the term consistent for well-defined.

Some sources do not hyphenate: well defined.

Examples

Less Than Relation on Congruence Modulo $6$

Let $x \mathrel {C_6} y$ be the equivalence relation defined on the natural numbers as congruence modulo $6$:

$x \mathrel {C_6} y \iff x \equiv y \pmod 6$

defined as:

$\forall x, y \in \N: x \equiv y \pmod 6 \iff \exists k, l, m \in \N, m < 6: 6 k + m = x \text { and } 6 l + m = y$

Let $\eqclass x {C_6}$ denote the equivalence class of $x$ under $C_6$.

Let $\N / {C_6}$ denote the quotient set of $\N$ by $C_6$.

Let us define the relation $L$ on $\N / {C_6}$ as follows:

$\tuple {\eqclass x {C_6}, \eqclass y {C_6} } \in L \iff \exists k, l, m, n \in \N, m < n < 6: x = 6 k + m, y = 6 l + n$

Then $L$ is a well-defined relation.

Also see

• Results about well-defined relations can be found here.

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions