Definition:Well-Defined/Relation

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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.


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