Inverse relation

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 LC, LT, 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.

Contents

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

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • inverse — in‧verse [ˌɪnˈvɜːs◂ ǁ ɜːrs◂] adjective in inverse proportion/​relation to something used for saying that one thing increases at the same rate as another related thing gets smaller: • Stocks moved in inverse relation to oil prices throughout the… …   Financial and business terms

  • Inverse (mathematics) — Inverse is the opposite of something. This word and its derivatives are used greatly in mathematics, as illustrated below. * Inverse element of an element x with respect to a binary operation * with identity element e is an element y such that x… …   Wikipedia

  • Inverse function — In mathematics, if fnof; is a function from A to B then an inverse function for fnof; is a function in the opposite direction, from B to A , with the property that a round trip (a composition) from A to B to A (or from B to A to B ) returns each… …   Wikipedia

  • Relation (mathematics) — This article sets out the set theoretic notion of relation. For a more elementary point of view, see binary relations and triadic relations. : For a more combinatorial viewpoint, see theory of relations. In mathematics, especially set theory, and …   Wikipedia

  • Relation (Mathematik) — Eine Relation ist allgemein eine Beziehung, die zwischen Dingen bestehen kann. Relationen im Sinne der Mathematik sind ausschließlich diejenigen Beziehungen, bei denen stets klar ist, ob sie bestehen oder nicht. Zwei Gegenstände können also nicht …   Deutsch Wikipedia

  • Relation (Mengentheorie) — Eine Relation ist allgemein eine Beziehung, die zwischen Dingen bestehen kann. Relationen im Sinne der Mathematik sind ausschließlich diejenigen Beziehungen, bei denen stets klar ist, ob sie bestehen oder nicht. Zwei Gegenstände können also nicht …   Deutsch Wikipedia

  • Inverse relationship — An inverse or negative relationship is a mathematical relationship in which one variable decreases as another increases. For example, there is an inverse relationship between education and unemployment that is, as education increases, the rate of …   Wikipedia

  • Relation ternaire interne — Une relation ternaire interne dans un ensemble associe des éléments de cet ensemble à des couples formés d’éléments de ce même ensemble. Sommaire 1 Définitions 2 Exemples 2.1 Propriétés …   Wikipédia en Français

  • inverse — [ ɛ̃vɛrs ] adj. et n. m. • 1611; envers XIIe; lat. inversus, de invertere « retourner » I ♦ Adj. 1 ♦ (Direction, ordre) Qui est exactement opposé, contraire. Dans l ordre inverse. Une relation inverse. Tourner dans le sens inve …   Encyclopédie Universelle

  • RELATION — Le concept de relation apparaît comme l’un des concepts fondamentaux du discours rationnel. Il semble lié à la pratique de l’analyse, qui constitue elle même l’un des aspects essentiels de la démarche discursive. L’analyse décompose les unités… …   Encyclopédie Universelle

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”