Converse relation

In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse 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 $L T$ is the relation defined so that $y L T x$ if and only if $x L y$ . In set-builder notation, $L T = {(y, x) \in Y \times X | (x, y) \in L$ }.

The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations as detailed below. As a unary operation, taking the converse (sometimes called conversion or transposition) commutes with the order-related operations of the calculus of relations, that is it commutes with union, intersection, and complement.

The converse relation is also called the or transpose relation— the latter in view of its similarity with the transpose of a matrix.^[1] It has also been called the opposite or dual of the original relation.^[2] Other notations for the converse relation include $L C, L -1, L ~$ or ${\breve {L}}$ or $L °$ or $L \lor$ .

Examples

For the usual (maybe strict or partial) order relations, the converse is the naively expected "opposite" order, for examples, ${\leq ^{\mathsf {T}}}={\geq },\quad {<^{\mathsf {T}}}={>}.$

A relation may be represented by a logical matrix such as

{\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}}.

Then the converse relation is represented by its transpose matrix:

{\begin{pmatrix}1&0&0&0\\1&1&0&0\\1&0&1&0\\1&1&0&1\end{pmatrix}}.

The converse of kinship relations are named: "A is a child of B" has converse "B is a parent of A". "A is a nephew or niece of B" has converse "B is an uncle or aunt of A". The relation "A is a sibling of B" is its own converse, since it is a symmetric relation.

Properties

In the monoid of binary endorelations on a set (with the binary operation on relations being the composition of relations), the converse relation does not satisfy the definition of an inverse from group theory, i.e. if L is an arbitrary relation on X, then $L\circ L^{\mathsf {T}}$ does not equal the identity relation on X in general. The converse relation does satisfy the (weaker) axioms of a semigroup with involution: $\left(L^{\mathsf {T}}\right)^{\mathsf {T}}=L$ and $\left(L\circ R\right)^{\mathsf {T}}=R^{\mathsf {T}}\circ L^{\mathsf {T}}$ .^[3]

Since one may generally consider relations between different sets (which form a category rather than a monoid, namely the category of relations Rel), in this context the converse relation conforms to the axioms of a dagger category (aka category with involution).^[3] A relation equal to its converse is a symmetric relation; in the language of dagger categories, it is self-adjoint.

Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations as sets), and actually an involutive quantale. Similarly, the category of heterogeneous relations, Rel is also an ordered category.^[3]

In the calculus of relations, conversion (the unary operation of taking the converse relation) commutes with other binary operations of union and intersection. Conversion also commutes with unary operation of complementation as well as with taking suprema and infima. Conversion is also compatible with the ordering of relations by inclusion.^[1]

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 converse is too.

Inverses

If I represents the identity relation, then a relation R may have an inverse as follows:

A relation R is called right-invertible if there exists a relation X with

R\circ X=I

, and left-invertible if there exists a Y with

Y\circ R=I

. Then X and Y are called the right and left inverse of R, respectively. Right- and left-invertible relations are called invertible. For invertible homogeneous relations all right and left inverses coincide; the notion inverse R^–1 is used. Then R^–1 = R^T holds.^[1]^:79

Converse relation of a function

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

The converse relation of a function $f:X\to Y$ is the relation $f^{{-1}}:Y\to X$ defined by $\operatorname {graph} \,f^{-1}=\left\{(y,x)\mid y=f(x)\right\}$ .

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.

For example, the function $f(x)=2x+2$ has the inverse function $f^{-1}(x)={\frac {x}{2}}-1$ .

However, the function $g(x) = x^2$ has the inverse relation $g^{-1}(x)=\pm x^{\frac {1}{2}}$ , which is not a function, being multi-valued.

References

1 2 3 Gunther Schmidt; Thomas Ströhlein (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Berlin Heidelberg. pp. 9–10. ISBN 978-3-642-77970-1.
↑ Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups. Kluwer Academic Publishers. p. 3. ISBN 978-1-4613-0267-4.
1 2 3 Joachim Lambek (2001). "Relations Old and New". In Ewa Orlowska, Andrzej Szalas. Relational Methods for Computer Science Applications. Springer Science & Business Media. pp. 135–146. ISBN 978-3-7908-1365-4.

Halmos, Paul R. (1974), Naive Set Theory, p. 40, ISBN 978-0-387-90092-6

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[R&G-1] 1 2 3 Gunther Schmidt; Thomas Ströhlein (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer Berlin Heidelberg. pp. 9–10. ISBN 978-3-642-77970-1.

[2] Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups. Kluwer Academic Publishers. p. 3. ISBN 978-1-4613-0267-4.

[Lambek2001-3] 1 2 3 Joachim Lambek (2001). "Relations Old and New". In Ewa Orlowska, Andrzej Szalas. Relational Methods for Computer Science Applications. Springer Science & Business Media. pp. 135–146. ISBN 978-3-7908-1365-4.