Cartesian product

In mathematics, the Cartesian product of two sets X and Y is the set of ordered pairs from X and Y. The product of any finite number of sets may be defined inductively.

The product of a general family of sets X_λ as λ ranges over a general index set Λ may be defined as the set of all functions x on Λ such that x(λ) is in X_λ for all λ in Λ. The Axiom of Choice is equivalent to stating that an element of such a product may always be taken.

References

• Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold, 24. 
• Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 12. ISBN 0-387-90441-7.