Inverse relation

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In mathematics, the inverse relation of a binary relation is the relation taken 'backwards', as in changing the relation 'child of' to 'parent of'. In formal terms, if

L : X \to Y is a binary relation with \operatorname{graph}\,L\subset X\times Y

then the inverse relation is

L^{-1} : Y \to X defined by y\,L^{-1}\,x\iff x\,L\,y ,

i.e. with \operatorname{graph}\,L^{-1} = \{(y, x)\in Y\times X\mid (x, y) \in \operatorname{graph}\, L\}.

The inverse relation is also called the converse relation and may be written as LC, LT (in view of its similarity with the transpose of a matrix), or \breve{L}.

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[edit] Properties

A relation equal to its inverse is a symmetric relation.

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomoushttp://en.wikipedia.org../../../../articles/b/i/n/Binary_relation.html#Relations_over_a_set, a partial order, total order, strict weak order, total preorderhttp://en.wikipedia.org../../../../articles/s/t/r/Strict_weak_order.html#Total_preorders (weak order), or an equivalence relation, its inverse is too.

However, if a relation is extendablehttp://en.wikipedia.org../../../../articles/b/i/n/Binary_relation.html#Relations_over_a_set, this need not be the case for the inverse.

[edit] Examples

For usual (maybe strict or partial) order relations, the converse is the naively expected "opposite" order, e.g. (\le)^{-1}=\ \ge ,~ (<)^{-1}=\ > , etc. (Parentheses would not be needed here but have been added for clarity.)

[edit] Inverse relation of a function

The inverse relation of a function f : X \to Y is the relation f^{-1} : Y \to X defined by \operatorname{graph}\, f^{-1} = \{(y, x) \mid y = f(x) \}. This is not necessarily a function: One necessary condition is that f be injective, since else f - 1 is multi-valued. This condition is sufficient for f - 1 being a partial function, and it is clear that f - 1 then is a (total) function if and only if f is surjective. In that case, i.e. if f is bijective, f - 1 may be called the inverse function of f.

[edit] See also

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