Converse relation information

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.

^ 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.
^ 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

[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

Converse relation information

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