In mathematics, a finitary relation has a finite number of "places". In set theory and logic, a relation is a property that assigns truth values to -tuples of individuals. Typically, the property describes a possible connection between the components of a -tuple. For a given set of -tuples, a truth value is assigned to each -tuple according to whether the property does or does not hold.An example of a ternary relation (i.e., between three individuals) is: "was introduced to by ", where is a 3-tuple of persons; for example, "Beatrice Wood was introduced to Henri-Pierre Roché by Marcel Duchamp" is true, while "Karl Marx was introduced to Friedrich Engels by Queen Victoria" is false.The variable giving the number of "places" in the relation, 3 for the above example, is a non-negative integer, called the relation's arity, adicity, or dimension. A relation with places is variously called a -ary, a -adic, or a -dimensional relation. Relations with a finite number of places are called finite-place or finitary relations. It is possible to generalize the concept to include infinitary relations between infinitudes of individuals, for example infinite sequences; however, in this article only finitary relations are discussed, which will from now on simply be called relations.Since there is only one 0-tuple, the so-called empty tuple ( ), there are only two zero-place relations: the one that always holds, and the one that never holds. They are sometimes useful for constructing the base case of an induction argument. One-place relations are called unary relations. For instance, any set (such as the collection of Nobel laureates) can be viewed as a collection of individuals having some property (such as that of having been awarded the Nobel prize). Two-place relations are called binary relations or, in the past, dyadic relations. Binary relations are very common, given the ubiquity of relations such as: Equality and inequality, denoted by signs such as '' and '' in statements like ''; Being a divisor of, denoted by the sign '' in statements like ''; Set membership, denoted by the sign '' in statements like ''.A -ary relation is a straightforward generalization of a binary relation.
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