Relation (database)

In database theory, a relation, as originally defined by E. F. Codd,^[1] is a set of tuples (d₁,d₂,...,d_n), where each element d_j is a member of D_j, a data domain. Codd's original definition notwithstanding, and contrary to the usual definition in mathematics, there is no ordering to the elements of the tuples of a relation.^[2][3] Instead, each element is termed an attribute value. An attribute is a name paired with a domain (nowadays more commonly referred to as a type or data type). An attribute value is an attribute name paired with an element of that attribute's domain, and a tuple is a set of attribute values in which no two distinct elements have the same name. Thus, in some accounts, a tuple is described as a function, mapping names to values.

A set of attributes in which no two distinct elements have the same name is called a heading. It follows from the above definitions that to every tuple there corresponds a unique heading, being the set of names from the tuple, paired with the domains from which the tuple elements' domains are taken. A set of tuples that all correspond to the same heading is called a body. A relation is thus a heading paired with a body, the heading of the relation being also the heading of each tuple in its body. The number of attributes constituting a heading is called the degree, which term also applies to tuples and relations. The term n-tuple refers to a tuple of degree n (n ≥ 0).

E. F. Codd used the term "relation" in its mathematical sense of a finitary relation, a set of tuples on some set of n sets S₁,S₂,....,S_n.^[4] Thus, an n-ary relation is interpreted, under the Closed-World Assumption, as the extension of some n-adic predicate: all and only those n-tuples whose values, substituted for corresponding free variables in the predicate, yield propositions that hold true, appear in the relation.

A heading paired with a set of constraints defined in terms of that heading is called a relation schema. A relation can thus be seen as an instantiation of a relation schema if it has the heading of that schema and it satisfies the applicable constraints.

Sometimes a relation schema is taken to include a name.^[5][6] A relational database definition (database schema, sometimes referred to as a relational schema) can thus be thought of as a collection of named relation schemas.^[7][8]

In implementations, the domain of each attribute is effectively a data type^[9] and a named relation schema is effectively a relation variable (relvar for short).

In SQL, a database language for relational databases, relations are represented by tables, where each row of a table represents a single tuple, and where the values of each attribute form a column.