Қатынастар: Ақиқат мәнін беретін топтамалар туралы теория

Property that assigns truth values to k tuples of individuals

In mathematics, a finitary relation over a sequence of sets X1, , Xn is a subset of the Cartesian product X1 × × Xn; that is, it is a set of n tuples (x1, , xn), each being a sequence of elements xi in the corresponding 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 that gives 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.

Nullary (0 ary) relations count only two members: the empty nullary relation, which never holds, and the universal nullary relation, which always holds. This is because there is only one 0 tuple, the empty tuple , and there are exactly two subsets of the (singleton) set of all 0 tuples. They are sometimes useful for constructing the base case of an induction argument.

Unary (1 ary) 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). Every nullary function is a unary relation.

Ternary (3 ary) relations include, for example, the binary functions, which relate two inputs and the output. All three of the domains of a homogeneous ternary relation are the same set.

The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relation in anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan and relations, see Merrill 1990). Charles Peirce, Gottlob Frege, Georg Cantor, Richard Dedekind and others advanced the theory of relations. Many of their ideas, especially on relations called orders, were summarized in The Principles of Mathematics (1903) where Bertrand Russell made free use of these results. In 1970, Edgar Codd proposed a relational model for databases, thus anticipating the development of data base management systems.