Cartesian product: Let A and B be two non-empty sets. Then the set of all possible ordered pairs (x, y) such that the first component x of the ordered pairs is an element of set A, and the second component y is an element of set B, is called the cartesian product of the sets A and B. It is denoted by A × B read as “A cross B”. 

A × B = {(a, b), a ∈ A and b ∈ B} 

Also, n(A × B) = n(A) × n(B) = pq if set A has p elements and set B has q elements.

Notes: 1. The cartesian product A × B is not the same as B × A. 

In A × B, the set A is named first so its elements will appear as the first components of the ordered pairs. 

In B × A, the set B is named first, so its elements will appear as the first components of the ordered pairs. 

2. If either A or B is a null set, then we define A × B to be a null set. 

If A = {a, b} and B = ϕ then A × B = ϕ 

3. If either A or B is an infinite set and the other is a non-empty set, then A × B is also an infinite set. 

4. If A and B are two non-empty sets having n-elements in common, then A × B and B × A have n² elements in common. 

5. If A = B, then A × B = A x A and is denoted by A². 

Examples: 

Ex. 1. If A = {a, b} and B = {1, 2, 3}, then 

A × B = {(a, 1), (b, 1), (a, 2), (b, 2), (a, 3), (b, 3)} 

B × A = {(1, a), (2, a), (3, a), (1, b), (2, b), (3, b)}. 

Ex. 2. If A = {2, 4, 6}, then 

A × A = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}.