Mathematical relation

In mathematics, an n-ary 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-ary tuple R=(X₁, ..., X_n, G(R)) where G(R) is a subset of X₁ × ... × X_n (the Cartesian product of these sets). G(R) is called the graph of R and, similar to the case of binary relation, R is often identified as 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 iff (x₁, ..., x_n) is in 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:

( x₁ , x₂ , ... ) ∈ R

R( x₁ , x₂ , ... )

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 called n-ary; for example "a 5-ary relation".