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| In [[mathematics]], a '''partial equivalence relation''' (often abbreviated as '''PER''') <math>R</math> on a set <math>X</math> is a relation that is ''[[symmetric relation|symmetric]]'' and ''[[transitive relation|transitive]]''. In other words, it holds for all <math>a, b, c \in X</math> that:
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| # if <math>a R b</math>, then <math>b R a</math> (symmetry)
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| # if <math>a R b</math> and <math>b R c</math>, then <math>a R c</math> (transitivity)
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| If <math>R</math> is also [[reflexive relation|reflexive]], then <math>R</math> is an [[equivalence relation]].
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| In a [[set-theoretic]] context, there is a simple structure to the general PER <math>R</math> on <math>X</math>: it is an equivalence relation on the subset <math>Y = \{ x \in X | x\,R\,x\} \subseteq X</math>. (<math>Y</math> is the subset of <math>X</math> such that in the [[Complement (set theory)|complement]] of <math>Y</math> (<math>X\setminus Y</math>) no element is related by <math>R</math> to any other.) By construction, <math>R</math> is reflexive on <math>Y</math> and therefore an equivalence relation on <math>Y</math>. Notice that <math>R</math> is actually only true on elements of <math>Y</math>: if <math>x R y</math>, then <math>y R x</math> by symmetry, so <math>x R x</math> and <math>y R y</math> by transitivity. Conversely, given a subset ''Y'' of ''X'', any equivalence relation on ''Y'' is automatically a PER on ''X''.
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| PERs are therefore used mainly in [[computer science]], [[type theory]] and [[constructive mathematics]], particularly to define [[setoid]]s, 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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| ==Examples==
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| A simple example of a PER that is ''not'' an equivalence relation is the empty relation <math>R=\emptyset</math> (unless <math>X=\emptyset</math>, in which case the empty relation ''is'' an equivalence relation (and is the only relation on <math>X</math>)).
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| ===Kernels of partial functions===
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| For another example of a PER, consider a set <math>A</math> and a partial function <math>f</math> that is defined on some elements of <math>A</math> but not all. Then the relation <math>\approx</math> defined by
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| : <math>x \approx y</math> if and only if <math>f</math> is defined at <math>x</math>, <math>f</math> is defined at <math>y</math>, and <math>f(x) = f(y)</math>
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| is a partial equivalence relation but not an equivalence relation. It possesses the symmetry and transitivity properties, but it is not reflexive since if <math>f(x)</math> is not defined then <math>x \not\approx x</math> — in fact, for such an <math>x</math> there is no <math>y \in A</math> such that <math>x \approx y</math>. (It follows immediately that the subset of <math>A</math> for which <math>\approx</math> is an equivalence relation is precisely the subset on which <math>f</math> is defined.)
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| ===Functions respecting equivalence relations===
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| Let ''X'' and ''Y'' be sets equipped with equivalence relations (or PERs) <math>\approx_X, \approx_Y</math>. For <math>f,g : X \to Y</math>, define <math>f \approx g</math> to mean:
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| : <math>\forall x_0 \; x_1, \quad x_0 \approx_X x_1 \Rightarrow f(x_0) \approx_Y g(x_1)</math> | |
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| then <math>f \approx f</math> means that ''f'' induces a well-defined function of the quotients <math>X / \approx_X \; \to \; Y /
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| \approx_Y</math>. Thus, the PER <math>\approx</math> captures both the idea of ''definedness'' on the quotients and of two functions inducing the same function on the quotient.
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| ==References==
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| * Mitchell, John C. ''[http://portal.acm.org/citation.cfm?id=237842 Foundations of programming languages.]'' MIT Press, 1996.
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| ==See also==
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| *[[Equivalence relation]]
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| *[[Binary relation]]
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| {{DEFAULTSORT:Partial Equivalence Relation}}
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| [[Category:Mathematical relations]]
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Greetings! I am Myrtle Shroyer. Years ago we moved to North Dakota. For many years he's been operating as a meter reader and it's some thing he truly appreciate. The favorite pastime for my children and me is to perform baseball but I haven't made a dime with it.
My blog :: std home test