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| − | A [[Binary relation|binary relation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420901.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420902.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420903.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420904.png" /> is the diagonal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420905.png" />. This means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420907.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420908.png" />, that is, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f0420909.png" /> there is at most one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209011.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209012.png" /> determines a function (perhaps not defined everywhere) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209013.png" />. When it satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209014.png" /> this function is well-defined everywhere and is one-to-one. | + | {{TEX|done}} |
| | + | 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. |
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| − | A functional relation is more generally defined as a binary relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209015.png" /> between sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209019.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042090/f04209020.png" />. | + | 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 $b=c$. |
Revision as of 20:41, 14 April 2014
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.
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=13901
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article