# Cartesian Product of Two Sets

Let A and B be two sets. The Cartesian product or cross product of A and B, denoted by A Ã— B, is the set of all ordered pairs (a, b) such that a âˆˆ A and b âˆˆ B.

That is, A Ã— B = {(a, b) |a âˆˆ A, b âˆˆ B}.

Thus, {1, 2, 3} Ã— {a, b} = {(1, a), (2, a), (3, a), (1, b), (2, b), (3, b)}

Note carefully that

- A Ã—B B Ã— A unless B = A.
- If n(A) stands for the number of elements in A, then n(A Ã— B) = n(B Ã— A) = n(A) Ã— n(B).

- If either of A or B is a null set, then A Ã— B is also a null set.
- If either of A or B is an infinite set, then A Ã— B is also an infinite set.
- A Ã—A Ã— A = {(a, b, c) | a, b, c âˆˆ A}. Here (a, b, c) is called an ordered triplet.