View source for Binary relation

{{TEX|done}}{{MSC|03}}

A two-place [[Predicate|predicate]] on a given set. The term is sometimes used to denote a subset of the set $A\times A$ of ordered pairs $(a,b)$ of elements of a given set $A$. A binary relation is a special case of a [[Relation|relation]]. Let $R\subseteq A\times A$. If $(a,b)\in R$, then one says that the element $a$ is in binary relation $R$ to the element $b$. An alternative notation for $(a,b)\in R$ is $aRb$.

The empty subset $\emptyset$ in $A\times A$ and the set $A\times A$ itself are called, respectively, the nil relation and the universal relation in the set $A$. The diagonal of the set $A\times A$, i.e. the set $\Delta=\{(a,a)\colon a\in A\}$, is the equality relation or the identity binary relation in $A$.

Let $R,R_1,R_2$ be binary relations in a set $A$. In addition to the set-theoretic operations of [[Union of sets|union]] $R_1\cup R_2$, [[Intersection of sets|intersection]] $R_1\cap R_2$, and negation or [[Relative complement|complementation]] $R'=(A\times A)\setminus R$, one has the inversion (also inverse, converse or transpose)

$$R^{-1}=\{(a,b)\colon(b,a)\in R\},$$

as well as the operation of multiplication (or [[composition]]):

$$R_1\circ R_2=\{(a,c) \in A\times A\colon(\exists b\in A)(aR_1b\text{ and }bR_2c)\}.$$

Multiplication of binary relations is associative, but as a rule not commutative.

A binary relation $R$ in $A$ is said to be 1) [[Reflexivity|reflexive]] if $\Delta\subseteq R$; 2) [[Transitive relation|transitive]] if $R\circ R\subseteq R$; 3) [[Symmetry (of a relation)|symmetric]] if $R^{-1}\subseteq R$; and 4) anti-symmetric if $R\cap R^{-1}\subseteq\Delta$. If a binary relation has some of the properties 1), 2), 3) or 4), the inverse relation $R^{-1}$ has these properties as well. The binary relation $R\subseteq A\times A$ is said to be [[Functional relation|functional]] if $R^{-1}\circ R\subseteq\Delta$.

The most important types of binary relations are [[equivalence]]s, order relations ([[totally ordered set|total]] and [[partially ordered set|partial]]), and [[functional relation]]s.

====Comments====
More generally, a binary relation $R$ may be defined between two sets $A$ and $B$ as a two-place predicate or realised as a subset of the Cartesian product $A \times B$.   As before we may speak of the union, intersection or negation of $R$ as a relation on $A \times B$.  The transpose $R^t$ is now a relation on $B \times A$.   Given relations $R_1$ on $A\times B$ and $R_2$ on $B\times C$, the composition $R_1 \circ R_2$ is a relation on $A \times C$:

$$R_1\circ R_2=\{(a,c) \in A \times C\colon(\exists b\in B)(aR_1b\text{ and }bR_2c)\}.$$

A relation $R$ on $A\times B$ is ''functional'' if for each $a \in A$ there is at most one $b \in B$ such that $a R b$.  Such a relation defines a (partial) function from $A$ to $B$: cf. [[Functional relation]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P. R. Halmos, ''Naive Set Theory'', Springer (1960) {{ISBN|0-387-90092-6}}</TD></TR>
</table>

====Comments====
The set $\mathcal{R}(A)$ of binary relations on $A$ form a [[monoid]] under composition of relations with the equality relation as [[identity element]] and the empty relation as a [[zero]]; indeed, it is a [[Baer semigroup]].

====References====
 <table>
<TR><TD valign="top">[b1]</TD> <TD valign="top"> T.S. Blyth  ''Lattices and Ordered Algebraic Structures''  Springer (2006) {{ISBN|184628127X}}</TD></TR>
<TR><TD valign="top">[b2]</TD> <TD valign="top">  R. Fraïssé, ''Theory of Relations'', Studies in Logic and the Foundations of Mathematics, Elsevier (2011) {{ISBN|0080960413}}</TD></TR>
</table>