# Cartesian Products of Sets in math

Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e.,

P × Q = { (p,q) : p ∈ P, q ∈ Q }

If either P or Q is the null set, then P × Q will also be empty set, i.e., P × Q = φ

## Keys points of Cartesian Products of Sets in math

• Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.
• If there are p elements in A and q elements in B, then there will be pq elements in A × B, i.e., if n(A) = p and n(B) = q, then n(A × B) = pq.
• If A and B are non-empty sets and either A or B is an infinite set, then so is A × B
• A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.

## Example of Cartesian Products of Sets in math

Example If P = {a, b, c} and Q = {r}, form the sets P × Q and Q × P

Solution By the definition of the cartesian product,

P × Q = {(a, r), (b, r), (c, r)} and Q × P = {(r, a), (r, b), (r, c)}

Example If P = {1, 2}, form the set P × P × P.

Solution We have, P× P× P = {(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1),
(2,2,2)}.