# Finitary relation

## Property that assigns truth values to k-tuples of individuals / From Wikipedia, the free encyclopedia

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In mathematics, a **finitary relation** over sets *X*_{1}, ..., *X*_{n} is a subset of the Cartesian product *X*_{1} × ⋯ × *X*_{n}; that is, it is a set of *n*-tuples (*x*_{1}, ..., *x*_{n}) consisting of elements *x*_{i} in *X*_{i}.[1][2][3] Typically, the relation describes a possible connection between the elements of an *n*-tuple. For example, the relation "*x* is divisible by *y* and *z*" consists of the set of 3-tuples such that when substituted to *x*, *y* and *z*, respectively, make the sentence true.

The non-negative integer *n* giving the number of "places" in the relation is called the *arity*, *adicity* or *degree* of the relation. A relation with *n* "places" is variously called an ** n-ary relation**, an

**or a**

*n*-adic relation**relation of degree**. Relations with a finite number of places are called

*n**finitary relations*(or simply

*relations*if the context is clear). It is also possible to generalize the concept to

*infinitary relations*with infinite sequences.[4]

An *n*-ary relation over sets *X*_{1}, ..., *X*_{n} is an element of the power set of *X*_{1} × ⋯ × *X*_{n}.

0-ary relations count only two members: the one that always holds, and the one that never holds. This is because there is only one 0-tuple, the empty tuple (). They are sometimes useful for constructing the base case of an induction argument.

Unary relations can be viewed as a collection of members (such as the collection of Nobel laureates) having some property (such as that of having been awarded the Nobel prize).

Binary relations are the most commonly studied form of finitary relations. When *X*_{1} = *X*_{2} it is called a homogeneous relation, for example:

- Equality and inequality, denoted by signs such as = and < in statements such as "5 < 12", or
- Divisibility, denoted by the sign | in statements such as "13|143".

Otherwise it is a heterogeneous relation, for example:

- Set membership, denoted by the sign ∈ in statements such as "1 ∈
**N**".