Inverse relation

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In logic and 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 $L C$ , $L T$ (in view of its similarity with the transpose of a matrix), or $\breve{L}$ .

A relation equal to its inverse is a symmetric relation.

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.)

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.

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Categories: Logic | Mathematical relations

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