Editing Tuple relational calculus (section)

=== Relational database ===

Since the calculus is a query language for [[relational database]]s we first have to define a relational database. The basic relational building block is the [[Data domain|domain]] (somewhat similar, but not equal to, a [[data type]]).  A [[tuple]] is a finite sequence of [[Attribute (computing)|attribute]]s, which are [[ordered pair]]s of domains and values. A [[relational model|relation]] is a set of (compatible) tuples.  Although these relational concepts are mathematically defined, those definitions map loosely to traditional database concepts.  A [[table (database)|table]] is an accepted visual representation of a relation; a tuple is similar to the concept of a [[row (database)|row]].

We first assume the existence of a set ''C'' of column names, examples of which are "name", "author", "address", etcetera. We define ''headers'' as finite subsets of ''C''. A ''relational database schema'' is defined as a [[tuple]] ''S'' = (''D'', ''R'', ''h'') where ''D'' is the domain of atomic values (see [[relational model]] for more on the notions of ''domain'' and ''atomic value''), ''R'' is a finite set of relation names, and

:''h'' : ''R'' → 2<sup>''C''</sup>

a [[function (mathematics)|function]] that associates a header with each relation name in ''R''. (Note that this is a simplification from the full relational model where there is more than one domain and a header is not just a set of column names but also maps these column names to a domain.) Given a domain ''D'' we define a ''tuple'' over ''D'' as a [[partial function]] that maps some column names to an atomic value in ''D''. An example would be (name : "Harry", age : 25).

:''t'' : ''C'' ⇸ ''D''

The set of all tuples over ''D'' is denoted as ''T''<sub>''D''</sub>. The subset of ''C'' for which a tuple ''t'' is defined is called the ''domain'' of ''t'' (not to be confused with the domain in the schema) and denoted as ''dom''(''t'').

Finally we define a ''relational database'' given a schema ''S'' = (''D'', ''R'', ''h'') as a function

:''db'' : ''R'' → 2<sup>''T''<sub>''D''</sub></sup>

that maps the relation names in ''R'' to finite subsets of ''T''<sub>''D''</sub>, such that for every relation name ''r'' in ''R'' and tuple ''t'' in ''db''(''r'') it holds that

:''dom''(''t'') = ''h''(''r'').

The latter requirement simply says that all the tuples in a relation should contain the same column names, namely those defined for it in the schema.