Converse relation

In mathematics, the converse 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 $L^{\operatorname {T} }$ is the relation defined so that $yL^{\operatorname {T} }x$ if and only if $xLy.$ In set-builder notation,

L^{\operatorname {T} }=\{(y,x)\in Y\times X:(x,y)\in L\}.

Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation^[1]^[2]^[3]^[4] is also called the transpose relation.^[5] It has also been called the opposite or dual of the original relation,^[6] the inverse of the original relation,^[7]^[8]^[9]^[10] or the reciprocal $L^{\circ }$ of the relation $L.$ ^[11]

Other notations for the converse relation include $L^{\operatorname {C} },L^{-1},{\breve {L}},L^{\circ },$ or $L^{\vee }.$ ^{[citation needed]}

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)^{[citation needed]} commutes with the order-related operations of the calculus of relations, that is it commutes with union, intersection, and complement.

Examples

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

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, that is, if $L$ is an arbitrary relation on $X,$ then $L\circ L^{\operatorname {T} }$ does not equal the identity relation on $X$ in general. The converse relation does satisfy the (weaker) axioms of a semigroup with involution: $\left(L^{\operatorname {T} }\right)^{\operatorname {T} }=L$ and $(L\circ R)^{\operatorname {T} }=R^{\operatorname {T} }\circ L^{\operatorname {T} }.$ ^[12]

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).^[12] 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.^[12]

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.^[5]

If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, connected, 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: $R$ is called

right-invertible: if there exists a relation $X,$ called a right inverse of $R,$ that satisfies $R\circ X=I.$
left-invertible: if there exists a relation $Y,$ called a left inverse of $R,$ that satisfies $Y\circ R=I.$
invertible: if it is both right-invertible and left-invertible.

For an invertible homogeneous relation $R,$ all right and left inverses coincide; this unique set is called its inverse and it is denoted by $R^{-1}.$ In this case, $R^{-1}=R^{\operatorname {T} }$ holds.^[5]^: 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 $f^{-1}\subseteq Y\times X$ defined by the $\operatorname {graph} \,f^{-1}=\{(y,x)\in Y\times X:y=f(x)\}.$

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, meaning 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 $g^{-1}(x)=\pm {\sqrt {x}},$ which is not a function, being multi-valued.

Composition with relation

Using composition of relations, the converse may be composed with the original relation. For example, the subset relation composed with its converse is always the universal relation:

∀A ∀B ∅ ⊂ A ∩B ⇔ A ⊃ ∅ ⊂ B ⇔ A ⊃ ⊂ B. Similarly,

For U = universe, A ∪ B ⊂ U ⇔ A ⊂ U ⊃ B ⇔ A ⊂ ⊃ B.

Now consider the set membership relation and its converse.

A\ni z\in B\Leftrightarrow z\in A\cap B\Leftrightarrow A\cap B\neq \emptyset .

Thus $A\ni \in B\Leftrightarrow A\cap B\neq \emptyset .$ The opposite composition $\in \ni$ is the universal relation.

The compositions are used to classify relations according to type: for a relation Q, when the identity relation on the range of Q contains Q^TQ, then Q is called univalent. When the identity relation on the domain of Q is contained in Q Q^T, then Q is called total. When Q is both univalent and total then it is a function. When Q^T is univalent, then Q is termed injective. When Q^T is total, Q is termed surjective.^[13]

If Q is univalent, then QQ^T is an equivalence relation on the domain of Q, see Transitive relation#Related properties.

References

^ Ernst Schröder, (1895), Algebra der Logik (Exakte Logik) Dritter Band, Algebra und Logik der Relative, Leibzig: B. G. Teubner via Internet Archive Seite 3 Konversion
^ Bertrand Russell (1903) Principles of Mathematics, page 97 via Internet Archive
^ C. I. Lewis (1918) A Survey of Symbolic Logic, page 273 via Internet Archive
^ Schmidt, Gunther (2010). Relational Mathematics. Cambridge: Cambridge University Press. p. 39. ISBN 978-0-521-76268-7.
^ ^a ^b ^c 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.
^ Daniel J. Velleman (2006). How to Prove It: A Structured Approach. Cambridge University Press. p. 173. ISBN 978-1-139-45097-3.
^ Shlomo Sternberg; Lynn Loomis (2014). Advanced Calculus. World Scientific Publishing Company. p. 9. ISBN 978-9814583930.
^ Rosen, Kenneth H. (2017). Handbook of discrete and combinatorial mathematics. Rosen, Kenneth H., Shier, Douglas R., Goddard, Wayne. (Second ed.). Boca Raton, FL. p. 43. ISBN 978-1-315-15648-4. OCLC 994604351.{{cite book}}: CS1 maint: location missing publisher (link)
^ Gerard O'Regan (2016): Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications ISBN 9783319445618
^ Peter J. Freyd & Andre Scedrov (1990) Categories, Allegories, page 79, North Holland ISBN 0-444-70368-3
^ ^a ^b ^c Joachim Lambek (2001). "Relations Old and New". In Ewa Orłowska; Andrzej Szalas (eds.). Relational Methods for Computer Science Applications. Springer Science & Business Media. pp. 135–146. ISBN 978-3-7908-1365-4.
^ Gunther Schmidt & Michael Winter (2018) Relational Topology, Springer Lecture Notes in Mathematics #2208, page 8, ISBN 978-3-319-74450-6

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

[1] Ernst Schröder, (1895), Algebra der Logik (Exakte Logik) Dritter Band, Algebra und Logik der Relative, Leibzig: B. G. Teubner via Internet Archive Seite 3 Konversion

[2] Bertrand Russell (1903) Principles of Mathematics, page 97 via Internet Archive

[3] C. I. Lewis (1918) A Survey of Symbolic Logic, page 273 via Internet Archive

[4] Schmidt, Gunther (2010). Relational Mathematics. Cambridge: Cambridge University Press. p. 39. ISBN 978-0-521-76268-7.

[R&G-5] 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.

[6] 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.

[7] Daniel J. Velleman (2006). How to Prove It: A Structured Approach. Cambridge University Press. p. 173. ISBN 978-1-139-45097-3.

[S&S-8] Shlomo Sternberg; Lynn Loomis (2014). Advanced Calculus. World Scientific Publishing Company. p. 9. ISBN 978-9814583930.

[9] Rosen, Kenneth H. (2017). Handbook of discrete and combinatorial mathematics. Rosen, Kenneth H., Shier, Douglas R., Goddard, Wayne. (Second ed.). Boca Raton, FL. p. 43. ISBN 978-1-315-15648-4. OCLC 994604351.{{cite book}}: CS1 maint: location missing publisher (link)

[rega2016-10] Gerard O'Regan (2016): Guide to Discrete Mathematics: An Accessible Introduction to the History, Theory, Logic and Applications ISBN 9783319445618

[11] Peter J. Freyd & Andre Scedrov (1990) Categories, Allegories, page 79, North Holland ISBN 0-444-70368-3

[Lambek2001-12] Joachim Lambek (2001). "Relations Old and New". In Ewa Orłowska; Andrzej Szalas (eds.). Relational Methods for Computer Science Applications. Springer Science & Business Media. pp. 135–146. ISBN 978-3-7908-1365-4.

[13] Gunther Schmidt & Michael Winter (2018) Relational Topology, Springer Lecture Notes in Mathematics #2208, page 8, ISBN 978-3-319-74450-6

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