Cartesian Product

Synonyms

Cross product

Definition

Given two relation instances R ₁, over set of attributes U ₁, and R ₂, over set of attributes U ₂ – with U ₁ and U ₂ disjoint – the cartesian product R ₁ × R ₂ returns a new relation, over set of attributes U ₁ ∪ U ₂, consisting of tuples {t|t(U ₁) ∈ R ₁ and t(U ₂) ∈ R ₂}. Here t(U) denotes the restriction of the tuple t to attributes in the set U.

Key Points

The cartesian product is an operator of the relational algebra which extends to relations the usual notion of cartesian product of sets.

Since the sets of attributes of the input relations are disjoint, in R ₁ × R ₂ each tuple of R ₁ is combined with each tuple of R ₂; moreover the arity of the output relation is the sum of the arities of R ₁ and R ₂.

As an example, consider a relation Students over attributes (student-number, student-name), containing tuples {(1001, Black), (1002, White)}, and a relation Courses over attributes (course-number, course-name), containing tuples {(EH1, Databases), (GH...