Inverse relation

In mathematics, the inverse relation of a binary relation is the relation that occurs when you switch the order of the elements in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms, if X and Y are sets and $L \subseteq X \times Y$ is a relation from X to Y then L ^{− 1} is the relation defined so that $y\,L^{-1}\,x$ if and only if $x\,L\,y$ (Halmos 1975, p. 40). In another way, $L^{-1} = \{(y, x) \in Y \times X \mid (x, y) \in L \}$.

The notation comes by analogy with that for an inverse function. Though many functions do not have an inverse; every relation does.

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 categories), and may be written as L^C, L^T, L^~ or $\breve{L}$.

Note that, despite the notation, the converse relation is not an inverse in the sense of composition of relations: $L \circ L^{-1} \neq \mathrm{id}$ in general.

Properties

A relation equal to its inverse is a symmetric relation (in the language of dagger categories, it is self-adjoint).

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, its inverse is too.

However, if a relation is extendable, this need not be the case for the inverse.

The operation of taking a relation to its inverse gives the category of relations Rel the structure of a dagger category.

The the set of all binary relations B(X) on a set X is a semigroup with involution with the involution being the mapping of a relation to its inverse relation.

Examples

For usual (maybe strict or partial) order relations, the converse is the naively expected "opposite" order, e.g. $\le^{-1}=\ \ge ,~ <^{-1}=\ >$, etc.

Inverse relation of a function

A function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inverse 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.

See also

• Bijection
• Function (mathematics)
• Inverse function
• Inverse relationship
• Relation (mathematics)

References

• Halmos, Paul R. (1974), Naive Set Theory, ISBN 978-0-387-90092-6