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.