Editing Tuple relational calculus (section)

=== Atoms ===

For the construction of the formulas we will assume an infinite set ''V'' of tuple variables. The formulas are defined given a database schema ''S'' = (''D'', ''R'', ''h'') and a partial function  ''type'' : ''V'' ⇸ 2<sup>''C''</sup>, called at ''type assignment'', that assigns headers to some tuple variables. We then define the ''set of [[atomic formula]]s'' ''A''[''S'',''type''] with the following rules:

# if ''v'' and ''w'' in ''V'', ''a'' in ''type''(''v'') and ''b'' in ''type''(''w'') then the formula ''v''.''a'' = ''w''.''b'' is in ''A''[''S'',''type''],
# if ''v'' in ''V'', ''a'' in ''type''(''v'') and ''k'' denotes a value in ''D'' then the formula ''v''.''a'' = ''k'' is in ''A''[''S'',''type''], and
# if ''v'' in ''V'', ''r'' in ''R'' and ''type''(''v'') = ''h''(''r'') then the formula ''r''(''v'') is in ''A''[''S'',''type''].

Examples of atoms are:

* (''t''.age = ''s''.age) — ''t'' has an age attribute and ''s'' has an age attribute with the same value
* (''t''.name = "Codd") — tuple ''t'' has a name attribute and its value is "Codd"
* Book(''t'') — tuple ''t'' is present in relation Book.

The formal semantics of such atoms is defined given a database ''db'' over ''S'' and a tuple variable binding ''val'' : ''V'' → ''T''<sub>''D''</sub> that maps tuple variables to tuples over the domain in ''S'':
#  ''v''.''a'' = ''w''.''b''  is true if and only if ''val''(''v'')(''a'') = ''val''(''w'')(''b'')
#  ''v''.''a'' = ''k''  is true if and only if ''val''(''v'')(''a'') = ''k''
#  ''r''(''v'')  is true if and only if ''val''(''v'') is in ''db''(''r'')