Cartesian Product
Let A and B be two non-empty sets. The cartesian product of A and B is denoted by A x B and is defined as the set of all ordered pairs (a, b).
where a ϵ A and b ϵ B.
Symbolically, A x B = {(a, b): a ϵ A and b ϵ B}
If there are three sets A, B and C and a ϵ A, b ϵ B, and c ϵ C, then we form an ordered triplet (a, b, c).
The set of all ordered triplets (a, b, c) is called the cartesian Product of three sets A, B and C
i.e., A x B x C = {(a, b, c): a ϵ A, b ϵ B, c ϵ C}
An ordered pair and ordered triplet are also called 2 – tuple and 3 – tuple, respectively
Example: Let A = {1, 2, 3} and B = {x, y}
A x B = {(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)}
and B x A = {(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)}
Note:
(i ) If A ≠ B, then A x B ≠ B x A
(ii) If A has p elements and B has q elements, then A x B has pq elements
(iii) If A = Ф and B = Ф then A x B = Ф
Cartesian Product of n sets A₁, A₂, A₃, … An is the set of all n – tuples (a₁, a₂, a₃, …, an), aᵢ ϵ Aᵢ, i = 1, 2, 3, …, n
It is denoted by A₁ x A₂ x A₃ x … x A n (or) \(\prod\limits_{i=1}^{n}{{{A}_{i}}}\).
Properties of Cartesian Product:
If A, B and C are three sets, then
(i) (a) A x (B ∪ C) = (A x B) ∪ (A x C)
(b) A x (B ∩ C) = (A x B) ∩ (A x C)
(ii) A x (B – C) = (A x B) – (A x C)
(iii) A x B = B x A
⇒ A = B
(iv) If A ⊆ B
⇒ A x A ⊆ (A x B) ∩ (B x A)
(v) If A ⊆ B ⇒ A x C ⊆ B x C
(vi) If A ⊆ B and C ⊆ D ⇒ A x C ⊆ B x D
(vii) If (A x B) ∩ (C x D) = (A ∩ C) x (B ∩ D)
(viii) If A x (B’ ∪ C’)’ = (A x B) ∩ (A x C)
(ix) A x (B’ ∩ C’)’ = (A x B) ∪ A x C).