# Cartesian product

The Carte­sian product of two sets $$A$$ and $$B,$$ de­noted $$A \times B,$$ is the set of all or­dered pairs $$(a, b)$$ such that $$a \in A$$ and $$b \in B.$$ For ex­am­ple, if $$\mathbb B \times \mathbb N$$ is the set of all pairs of a boolean with a nat­u­ral num­ber, and it con­tains el­e­ments like (true, 0), (false, 17), (true, 17), (true, 100), and (false, 101).

Carte­sian product are of­ten referred to as just “prod­ucts.”

Carte­sian prod­ucts can be con­structed from more than two sets, for ex­am­ple, $$\mathbb B^3 = \mathbb B \times \mathbb B \times \mathbb B$$ is the set of all boolean 3-tu­ples. (The $$\times$$ op­er­a­tor is as­so­ci­a­tive, so we don’t need to write paren­the­sis when us­ing it on a whole chain of sets.) A product of $$n$$ sets is called an $$n$$-ary product.