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Joel Brennan (talk | contribs) m →Relations: wording |
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[[Relation (mathematics)|Relations]] are sets whose members are all [[ordered pair]]s. Where possible, a relation <math>R</math> (understood as a [[binary predicate]]) is implemented as <math>\{(x,y) \mid x R y\}</math> (which may be written as <math>\{z \mid \pi_1(z) R \pi_2(z)\}</math>). When <math>R</math> is a relation, the notation <math>xRy</math> means <math>\left(x, y\right) \in R</math>.
In
to be sets (but may be harmlessly reified as [[proper class]]es). In
need to have the same type (because they appear as projections of the same pair), but also
successive types (because <math>x</math> is considered as an element of <math>y</math>).
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The '''___domain''' of <math>R</math> is the set <math>\left\{x : \exists y \left(xRy\right)\right\}</math>.
The '''range''' of <math>R</math> is the ___domain of the converse of <math>R</math>. That is, the set <math>\left\{y : \exists x \left(xRy\right)\right\}</math>.
The '''field''' of <math>R</math> is the [[union (set theory)|union]] of the ___domain and range of <math>R</math>.
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The '''[[relation composition|relative product]]''' <math>R|S</math> of <math>R</math> and <math>S</math> is the relation <math>\left\{\left(x, z\right) : \exists y\,\left(xRy \wedge ySz\right)\right\}</math>.
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Notice that with our formal definition of a binary relation, the range and codomain of a relation are not distinguished
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=== Properties and kinds of relations ===
*'''[[Reflexive relation|Reflexive]]''' if <math>xRx</math> for every <math>x</math> in the field of <math>R</math>.
* '''[[Symmetric relation|Symmetric]]''' if <math>\forall x, y \,(xRy \to yRx)</math>.
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* '''Extensional''' if for every <math>x, y</math> in the field of <math>R</math>, <math>x = y</math> if and only if <math>x</math> and <math>y</math> have the same preimage under <math>R</math>.
Relations having certain combinations of the above properties have standard names. A binary relation <math>R</math> is:
* An '''[[equivalence relation]]''' if <math>R</math> is reflexive, symmetric, and transitive.
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