Relational Algebra - Primitive Operations

Primitive Operations

As in any algebra, some operators are primitive and the others are derived in terms of the primitive ones. It is useful if the choice of primitive operators parallels the usual choice of primitive logical operators.

Five primitive operators of Codd's algebra are the selection, the projection, the Cartesian product (also called the cross product or cross join), the set union, and the set difference. Another operator, rename was not noted by Codd, but the need for it is shown by the inventors of ISBL. These six operators are fundamental in the sense that omitting any one of them causes a loss of expressive power. Many other operators have been defined in terms of these six. Among the most important are set intersection, division, and the natural join. In fact ISBL made a compelling case for replacing the Cartesian product with the natural join, of which the Cartesian product is a degenerate case.

Altogether, the operators of relational algebra have identical expressive power to that of domain relational calculus or tuple relational calculus. However, for the reasons given in section Introduction, relational algebra is less expressive than first-order predicate calculus without function symbols. Relational algebra corresponds to a subset of first-order logic, namely Horn clauses without recursion and negation.