# Cartesian Product

## 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).