Relation (mathematics)

In mathematics, an n-ary relation (or n-place relation or often simply relation) is a generalization of binary relations such as "=" and "<" which occur in statements such as "5 < 6" or "2 + 2 = 4". It is the fundamental notion in the relational model for databases.

Formally, a relation over the sets X₁, ..., X_n is an (n + 1)-tuple R=(X₁, ..., X_n, G(R)) where G(R) is a subset of X₁ × ... × X_n (the Cartesian product of these sets). If X=X₁=X₂=...=X_n, R is simply called a relation over X. G(R) is called the graph of R and, similar to the case of binary relations, R is often identified with its graph.

An n-ary predicate is a truth-valued function of n variables.

Because a relation as above defines uniquely an n-ary predicate that holds for x₁, ..., x_n if (x₁, ..., x_n) is in G(R), and vice versa, the relation and the predicate are often denoted with the same symbol. So, for example, the following two statements are considered to be equivalent:

$(x_1,x_2,\dotsb)\in G(R)$

$R(x_1,x_2,\dotsb)$

Relations are classified according to the number of sets in the Cartesian product; in other words the number of terms in the expression:

unary relation: R(x)
binary relation: R(x, y) or x R y
ternary relation: R(x, y, z)
quarternary relation: R(x, y, z, w)

Relations with more than 4 terms are usually called n-ary; for example "a 5-ary relation".

Categories: Set theory