# What is Cartesian Product of Sets – Definition & Example

Here you will learn what is cartesian product of sets and what is relation and inverse relation with example.

Let’s begin –

## Cartesian Product of Sets

The cartesian product of two sets A, B is a non-void set of all ordered pair (a,b),

where a $$\in$$ A and b $$\in$$ B. This is denoted by A $$\times$$ B.

$$\therefore$$   A $$\times$$ B = {(a,b) $$\forall$$ a $$\in$$ A and b $$\in$$ B}

e.g.   A = {1,2}, B = {a,b}

A $$\times$$ B = {(1,a), (1,b), (2,a), (2,b)}

Note :

(i)  A $$\times$$ B $$\ne$$ B $$\times$$ A    (Non-commutative)

(ii) n(A $$\times$$ B) = n(A)n(B) and n(P(A $$\times$$ B)) = $$2^{n(A)n(B)}$$

(iii) A = $$\phi$$ and B = $$\phi$$ $$\iff$$ A $$\times$$ B = $$\phi$$

(iv) If A and B are two non-empty sets having n elements in common, then (A $$\times$$ B) and (B $$\times$$ A) have $$n^2$$ elements in common

(v) A $$\times$$ (B $$\cup$$ C) = (A $$\times$$ B) $$\cup$$ (A $$\times$$ C)

(vi) A $$\times$$ (B $$\cap$$ C) = (A $$\times$$ B) $$\cap$$ (A $$\times$$ C)

(vii) A $$\times$$ (B – C) = (A $$\times$$ B) – (A $$\times$$ C)

## Relation

Every non-zero subset of A $$\times$$ B defined a relation from set A to set B.<br>If R is relation from A $$\rightarrow$$ B

R : {(a,b) | (a,b) $$\in$$ A $$\times$$ B and a R b}

Let A and B be two non empty sets and R : A $$\rightarrow$$ B be a relation such that R : {(a,b) | (a,b) $$\in$$ R a $$\in$$ A and b $$\in$$ B}

(i) ‘b’ is called image of ‘a’ under R.

(ii) ‘a’ is called pre-image of ‘b’ under R.

(iii) Domain of R : Collection of all elements of A which has a image in B.

(iv) Range of R : Collection of all elements of B which has a pre-image in A.

Note :

(1) It is not necessary that each and every element of set A has a image in set B and each and every element of set B has preimage in set A.

(2) Elements of set A having image in B is not necessarily unique.

(3) Basically relation is the number of subsets of A $$\times$$ B

Number of relations = no. of ways of selecting a non-zero subset of A $$\times$$ B

= $$^{mn}C_1$$+ $$^{mn}C_2$$ + …….. + $$^{mn}C_{mn}$$ = $$2^{mn} – 1$$

Total number of relation = $$2^{mn}$$(including void relation)

Example : If A = {1, 3, 5, 7}, B = {2, 4, 6, 8}
Relation is aRb $$\implies$$ a > b, a $$\in$$ A, a $$\in$$ B

Solution : R = {(3, 2), (5, 2), (5, 4), (7, 2), (7, 4), (7, 6)}
Domain = {3, 5, 7}
Range = {2, 4, 6}