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

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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 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)}.