Relation (database)

In <a href="/facts/Database/VYcpkevb">database</a> theory, a relation, as originally defined by <a href="/facts/Edgar_F._Codd/QCSbPYkK">E. F. Codd</a>, is a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of <a href="/facts/Tuple/BI1HIxL8">tuples</a> (d1,d2,...,dn), where each element dj is a member of Dj, a <a href="/facts/Data_domain/4oDqjYsd">data domain</a>. 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. 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 <a href="/facts/Data_type/fcx5hgVL">data type</a>). 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 <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>, 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 <a href="/facts/Tuple/BI1HIxL8">tuples</a> 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).
<a href="/facts/Edgar_F._Codd/QCSbPYkK">E. F. Codd</a> used the term "relation" in its mathematical sense of a <a href="/facts/Finitary_relation/JkNnVYNj">finitary relation</a>, a set of tuples on some set of n sets S1,S2,....,Sn. Thus, an n-ary relation is interpreted, under the <a href="/facts/Closed-world_assumption/Fnoe5m4F">Closed-World Assumption</a>, as the extension of some n-adic <a href="/facts/Predicate_(mathematical_logic)/5WazAvKF">predicate</a>: 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. A relational database definition (<a href="/facts/Database_schema/DBq8fooY">database schema</a>, sometimes referred to as a relational schema) can thus be thought of as a collection of named <a href="/facts/Relation_schema/Sj67AUyD">relation schemas</a>.
In implementations, the domain of each attribute is effectively a <a href="/facts/Data_type/fcx5hgVL">data type</a> and a named relation schema is effectively a <a href="/facts/Relvar/egcFY101">relation variable</a> (relvar for short).
In <a href="/facts/SQL/UjQLJcYm">SQL</a>, a <a href="/facts/Database_language/Sa6K4xXM">database language</a> for relational databases, relations are represented by <a href="/facts/Table_(database)/5INvK1Vd">tables</a>, where each row of a table represents a single tuple, and where the values of each attribute form a column.

Relation (database) open-in-new

Relation (database)