Cartesian product of sets

Cartesian product of sets A and B is denoted by A x B.

Set of all ordered pairs (a, b)of elements a∈ A, b ∈B then cartesian product A x B is {(a, b): a ∈A, b ∈ B}

Example – Let A = {1, 2, 3} and B = {4, 5}. Find A x B and B x A and show that A x B ≠ B x A.
Solution: AxB = {(1, 4) (1, 5) (2, 4) (2, 5) (3, 4) (3, 5)} and B x A = {(4, 1) (4, 2) (4, 3) (5, 1) (5, 2) (5, 3)}
From the above, we can see that (1, 4) ∈ A x B but (1, 4) ∉ A x B. So, A x B ≠ B x A.
Remarks:-
1. A ≠ ∅ or B = ∅ Then, A x B = ∅
2. A ≠ ∅ and B ≠ ∅ then A x B ≠ ∅
3.If the set A has ‘m’ elements and the set B has ‘n’ elements, then A x B has mn elements.
4. If A≠ ∅ and B ≠ ∅ wither A or B is an infinite set, so is A x B
5. If A = B then A x B = A²
6. If A, B and C are three sets then (a, b, c) where a A∈, b∈B and c x c then A x B x C = {(a, b, c): a ∈A, b∈B c∈c}

Examples on Cartesian product of sets

1) Let A = {a,b,c} and B= {p,q}. Find the cartesian product of sets A and B.(i) A X B (ii) B X A (iii) A x A (iv) B X B
Solution :
(i) A X B = { (a,p),(a,q)(b,p)(b,q),(c,p),(c,q)}
(ii)B X A = { (p,a),(p,b)(p,c)(q,a),(q,b),(q,c)}
(iii) A X A = { (a,a),(a,b)(a,c)(b,a),(b,b),(b,c),(c,a), (c,b),(c,c)}
(iv) B x B = { (p,p),(p,q)(q,p)(q,q)}

2) Let A = {1, 2, 3}, B = {3, 4} C = {4, 5, 6} Find A x (B ∩ C)
Solution : A = {1, 2, 3}
B = {3, 4}
C = {4, 5, 6}
So, B ∩ C = {4}
Now, A x (B ∩C) = {(1, 4) (2, 4) (3, 4)}

3) Find x and y if (x + 2, 4) = (5, 2x +y)
Solution: By definition of equal ordered x + 2 = 5
∴ x = 5 – 2
x = 3
2x + y = 4
2 (3) + y = 4
6 + y = 4
y = 4 – 6
y = -2
So, x = 3 and y = -2

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