Relation

A subset of a finite Cartesian power $A^n=A\times \cdots \times A$ of a given set $A$, i.e. a set of tuples $(a_1,\dots ,a_n)$ of $n$ elements of $A$.

A subset $R\subseteq A^n$ is called an $n$-place, or an $n$-ary, relation on $A$. The number $n$ is called the rank, or type, of the relation $R$. The notation $R(a_1,\dots ,a_n)$ signifies that $(a_1,\dots ,a_n)\in R$.

One-place relations are called properties. Two-place relations are called binary relations, three-place relations are called ternary, etc.

The set $A^n$ and the empty subset $\mathrm{\varnothing }$ in $R^n$ are called, respectively, the universal relation and the zero relation of rank $n$ on $A$. The diagonal of the set $A^n$, i.e. the set $$\mathrm{\Delta }=\{(a,a,\dots ,a):a\in A\}$$ is called the equality relation on $A$.

If $R$ and $S$ are $n$-place relations on $A$, then the following subsets of $A^n$ will also be $n$-place relations on $A$: $$R\cap SR\cup S,R^{\prime }=A^n\setminus RR\setminus S.$$

The set of all $n$-ary relations on $A$ is a Boolean algebra relative to the operations $\cup$, $\cap$, ${}^{\prime }$. An $(n+1)$-place relation $F$ on $A$ is called functional if for any elements $a_1,\dots ,a_n$, $a,b$, from $A$ it follows from $F(a_1,\dots ,a_n,a)$ and $F(a_1,\dots ,a_n,b)$ that $a=b$.

See also Binary relation; Correspondence; Predicate.

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