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 (x1, ..., xn) 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 (or simply relations if the context is clear). 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. 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 X1 = X2 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". Consider the ternary relation R "x thinks that y likes z" over the set of people P = {Alice, Bob, Charles, Denise}, defined by: R = {(Alice, Bob, Denise), (Charles, Alice, Bob), (Charles, Charles, Alice), (Denise, Denise, Denise)}.

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Related concepts (17)
Order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.
Algebraic structure
In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures.
Arity
In logic, mathematics, and computer science, arity (ˈærᵻti) is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and philosophy, arity may also be called adicity and degree. In linguistics, it is usually named valency. In general, functions or operators with a given arity follow the naming conventions of n-based numeral systems, such as binary and hexadecimal.
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