Difference between revisions of "Functional relation"
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A [[Binary relation|binary relation]] $R$ on a set $A$ satisfying $R^{-1}\circ R\subseteq\Delta$, where $\Delta$ is the diagonal of $A$. This means that $(a,b)\in R$ and $(a,c)\in R$ imply that $b=c$, that is, for each $a\in A$ there is at most one $b\in A$ such that $(a,b)\in R$. Thus, $R$ determines a function (perhaps not defined everywhere) on $A$. When it satisfies $R^{-1}\circ R=\Delta$ this function is well-defined everywhere and is one-to-one. | A [[Binary relation|binary relation]] $R$ on a set $A$ satisfying $R^{-1}\circ R\subseteq\Delta$, where $\Delta$ is the diagonal of $A$. This means that $(a,b)\in R$ and $(a,c)\in R$ imply that $b=c$, that is, for each $a\in A$ there is at most one $b\in A$ such that $(a,b)\in R$. Thus, $R$ determines a function (perhaps not defined everywhere) on $A$. When it satisfies $R^{-1}\circ R=\Delta$ this function is well-defined everywhere and is one-to-one. | ||
Revision as of 17:33, 23 November 2014
2020 Mathematics Subject Classification: Primary: 03-XX [MSN][ZBL]
A binary relation $R$ on a set $A$ satisfying $R^{-1}\circ R\subseteq\Delta$, where $\Delta$ is the diagonal of $A$. This means that $(a,b)\in R$ and $(a,c)\in R$ imply that $b=c$, that is, for each $a\in A$ there is at most one $b\in A$ such that $(a,b)\in R$. Thus, $R$ determines a function (perhaps not defined everywhere) on $A$. When it satisfies $R^{-1}\circ R=\Delta$ this function is well-defined everywhere and is one-to-one.
Comments
A functional relation is more generally defined as a binary relation $R\subset A\times B$ between sets $A$ and $B$ such that $(a,b)\in R$ and $(a,c)\in R$ imply $b=c$.
How to Cite This Entry:
Functional relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_relation&oldid=31682
Functional relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_relation&oldid=31682
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article