Cartesian Product

Synonyms

Cross product

Definition

Given two relation instances R₁, over set of attributes U₁, and R₂, over set of attributes U₂ – with U₁ and U₂ disjoint – the Cartesian product R₁ × R₂ returns a new relation, over set of attributes U₁ ∪ U₂, consisting of tuples {t|t(U₁) ∈ R₁ and t(U₂) ∈ R₂}. 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 R₁ × R₂, each tuple of R₁ is combined with each tuple of R₂; moreover the arity of the output relation is the sum of the arities of R₁ and R₂.

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), (GH5, Logic)}. Then Students...