Mathematics for Databases

A formal description of a database consists of the description of the relations (tables) of the database together with the constraints that must hold on the database. Furthermore the contents of a database can be retrieved using queries. These constraints and queries for databases can very well be formalized. A formal description of a constraint or a query is necessary to describe the constraint or query unambiguously. In other words, a formal description leads to one and only one meaning of the constraint or query. To describe constraints and queries in a formal way we use predicate logic, set theory and tuple relational calculus. The tuple relational calculus is a calculus based on the use of tuple variables. A tuple variable is a variable that ranges over a named relation (i.e. a set of tuples of a relation). This paper describes the use of the relational calculus for databases. A description of the formal notation is given as well as a mapping of these expressions to SQL.