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