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In mathematics, a finitary relation over sets X1,..., Xn is a subset of the Cartesian product X1 × ⋯ × Xn; that is, it is a set of n-tuples consisting of elements xi in Xi. 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 n-adic relation or a relation of degree n. Relations with a finite number of places are called finitary relations. It is also possible to generalize the concept to infinitary relations with infinite sequences.
An n-ary relation over sets X1,..., Xn is an element of the power set of X1 × ⋯ × Xn.
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.