# Binary relation

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Binary relationбинарное отношение.

$\,R$ is a binary relation on $\,V$ if $R \subseteq V \times V$. $\,R$ is reflexive on $\,V$ if for all $v \in V$ $(v,v) \in R$ and irreflexive otherwise. $\,R$ is transitive on $\,V$, if for all $u, v, w \in V$ $(u,v) \in R$ and $(v,w) \in R$ implies $(u,w) \in R$. $\,R$ is symmetric, if $(v,w) \in R$ implies $(w,v) \in R$, and antisymmetric on $V$, if for all $u, v \in V$ $(u,v) \in R$ and $(v,u) \in R$ implies $\,u = v$.

The inverse of a relation $\,R$, denoted by $\,R^{-1}$, is obtained by reversing each of the pairs belonging to $\,R$, so that $\,aR^{-1}b$ iff $\,bRa$. Let $\,R^{U}$ denote the union and $\,R^{I}$ the intersection of a collection of relations $\{R_{k}: \; k \in {\mathcal C}\}$ in $\,S$, where ${\mathcal C}$ is some nonempty index set. Then $\,aR^{U}b$ iff $\,aR_{k}b$ for some $\,k$ in ${\mathcal C}$, and $\,aR^{I}b$ iff $\,aR_{k}b$ for each $\,k$ in ${\mathcal C}$.