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| In [[mathematics]], the '''inverse relation''' of a [[binary relation]] is the relation that occurs when the order of the elements is switched in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms, if <math> X \text{ and } Y</math> are sets and <math>L \subseteq X \times Y</math> is a relation
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| from ''X'' to ''Y'' then <math>L^{-1}</math> is the relation defined so that <math>y\,L^{-1}\,x</math> if and only if <math>x\,L\,y</math> (Halmos 1975, p. 40). In another way, <math>L^{-1} = \{(y, x) \in Y \times X \mid (x, y) \in L \}</math>.
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| The notation comes by analogy with that for an [[inverse function]]. Though many functions do not have an inverse; every relation does. | |
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| The '''inverse relation''' is also called the '''converse relation''' or '''transpose relation''' (in view of its similarity with the [[transpose]] of a matrix: these are the most familiar examples of [[dagger category|dagger categories]]), and may be written as ''L''<sup>''C''</sup>, ''L''<sup>''T''</sup>, ''L''<sup>~</sup> or <math>\breve{L}</math>.
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| Note that, despite the notation, the converse relation is ''not'' an inverse in the sense of [[composition of relations]]: <math>L \circ L^{-1} \neq \mathrm{id}</math> in general.
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| ==Properties==
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| A relation equal to its inverse is a [[symmetric relation]] (in the language of [[dagger category|dagger categories]], it is '''self-adjoint''').
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| If a relation is [[reflexive relation|reflexive]], [[irreflexive relation|irreflexive]], [[symmetric relation|symmetric]], [[antisymmetric relation|antisymmetric]], [[asymmetric relation|asymmetric]], [[transitive relation|transitive]], [[total relation|total]], [[Binary_relation#Relations_over_a_set|trichotomous]], a [[partial order]], [[total order]], [[strict weak order]], [[Strict_weak_order#Total_preorders|total preorder]] (weak order), or an [[equivalence relation]], its inverse is too.
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| However, if a relation is [[Binary_relation#Relations_over_a_set|extendable]], this need not be the case for the inverse.
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| The operation of taking a relation to its inverse gives the [[category of relations]] '''Rel''' the structure of a [[dagger category]].
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| The set of all [[binary relation]]s '''''B'''''(X) on a set X is a [[semigroup with involution]] with the involution being the mapping of a relation to its inverse relation.
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| ==Examples==
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| For usual (maybe strict or partial) [[order relation]]s, the converse is the naively expected "opposite" order, e.g. <math> \le^{-1}=\ \ge ,~ <^{-1}=\ > </math>, etc.
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| ==Inverse relation of a function==
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| A function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inverse function.
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| The inverse relation of a [[function (mathematics)|function]] <math>f : X \to Y</math> is the relation <math>f^{-1} : Y \to X</math> defined by <math>\operatorname{graph}\, f^{-1} = \{(y, x) \mid y = f(x) \}</math>.
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| This is not necessarily a function: One necessary condition is that ''f'' be [[injective]], since else <math>f^{-1}</math> is [[multi-valued]]. This condition is sufficient for <math>f^{-1}</math> being a [[partial function]], and it is clear that <math>f^{-1}</math> then is a (total) function [[if and only if]] ''f'' is [[surjective]].
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| In that case, i.e. if ''f'' is [[bijective]], <math>f^{-1}</math> may be called the '''[[inverse function]]''' of ''f''.
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| ==See also==
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| * [[Bijection]]
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| * [[Function (mathematics)]]
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| * [[Inverse function]]
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| * [[Relation (mathematics)]]
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| * [[Transpose graph]]
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| == References ==
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| * {{Citation | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | title=[[Naive Set Theory (book)|Naive Set Theory]] | isbn=978-0-387-90092-6 | year=1974}}
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| [[Category:Mathematical logic]]
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| [[Category:Mathematical relations]]
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