Relational Model - Set-theoretic Formulation

Set-theoretic Formulation

Basic notions in the relational model are relation names and attribute names. We will represent these as strings such as "Person" and "name" and we will usually use the variables and to range over them. Another basic notion is the set of atomic values that contains values such as numbers and strings.

Our first definition concerns the notion of tuple, which formalizes the notion of row or record in a table:

Tuple

A tuple is a partial function from attribute names to atomic values.

Header

A header is a finite set of attribute names.

Projection

The projection of a tuple on a finite set of attributes is .

The next definition defines relation which formalizes the contents of a table as it is defined in the relational model.

Relation

A relation is a tuple with, the header, and, the body, a set of tuples that all have the domain .

Such a relation closely corresponds to what is usually called the extension of a predicate in first-order logic except that here we identify the places in the predicate with attribute names. Usually in the relational model a database schema is said to consist of a set of relation names, the headers that are associated with these names and the constraints that should hold for every instance of the database schema.

Relation universe

A relation universe over a header is a non-empty set of relations with header .

Relation schema

A relation schema consists of a header and a predicate that is defined for all relations with header . A relation satisfies a relation schema if it has header and satisfies .