Partial equivalence relation: Information from Answers.com

In mathematics, a partial equivalence relation (often abbreviated as PER) $R$ on a set $X$ is a relation that is symmetric and transitive. In other words, it holds for all $a, b, c \in X$ that:

if $a R b$ , then $b R a$ (symmetry)
if $a R b$ and $b R c$ , then $a R c$ (transitivity)

If $R$ is also reflexive, then $R$ is an equivalence relation.

A simple example of a PER that is not an equivalence relation is the empty relation $R=\emptyset$ (unless $X=\emptyset$ , in which case the empty relation is an equivalence relation (and is the only relation on $X$ )).

There is a simple structure to the general PER $R$ on $X$ : it is an equivalence relation on the subset $Y = \{ x \in X | x\,R\,x\} \subseteq X$ . ( $Y$ is the subset of $X$ such that in the complement of $Y$ ( $X\setminus Y$ ) no element is related by $R$ to any other.) By construction, $R$ is reflexive on $Y$ and therefore an equivalence relation on $Y$ . Notice that $R$ is actually only defined on elements of $Y$ : if $z \notin Y$ , then there is no $w \in X$ for which $z R w$ ; if there were, then by symmetry we would have $w R z$ , which would imply $z R z$ by transitivity, which would contradict $z \notin Y$ .

Example

For an example of a PER, consider a set $A$ and a partial function $f$ that is defined on some elements of $A$ but not all. Then the relation $\approx$ defined by

$x \approx y$ if and only if

f

is defined at

x

f

is defined at

y

, and

f (x) = f (y)

is a partial equivalence relation but not an equivalence relation. It possesses the symmetry and transitivity properties, but it is not reflexive since if $f (x)$ is not defined then $x \not\approx x$ — in fact, for such an $x$ there is no $y \in A$ such that $x \approx y$ . (It follows immediately that the subset of $A$ for which $\approx$ is an equivalence relation is precisely the subset on which $f$ is defined.)

Uses

PERs are used mainly in computer science, particularly in type theory. It is also used in constructive mathematics to define setoids, sometimes called partial setoids. The action of forming one from a type and a PER is analogous to the operations of subset and quotient in classical set-theoretic mathematics.

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Partial equivalence relation

Example

Uses

See also