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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, ..., Xn is an n+1-ary tuple R=(X1,
..., Xn, G(R)) where G(R) is a subset of X1 × ... × Xn (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, ..., xn iff (x1, ..., xn) 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".