This chapter deals with establishing binary relation between elements of one set and elements of another set according to some particular rule of relationship.


The Cartesian product of two sets `A, B` is a non-void set of all ordered pairs `(a, b)`, where `a in A` and `b in B`. This is denoted by `Axx B`.

`:. A xx B={(a,b)}|a in A` and `b in B}`

e.g. `A={1,2}, B{a,b}`

`A xx B={(1,a),(1,b),(2,a),(2,b)}`


Highlights :

Let `A` and `B` be two non empty sets and `R : A -> 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` or Set of all fust entries in `A xx B`.

(iv) Range of `R :` Collection of all elements of `B` which has a pre-image in `A` or Set of all second entries in `A xx B`.


`text(Important Theorems on Cartesian Product :)` If A, Band Care three sets, then

(i) `A xx (B uu C)= (A xx B) uu (A xx C)`
(ii) `Axx (B nn C)= (Axx B) nn (Axx C)`
(iii) `A xx (B - C)= (A xx B) - (A xx C)`
(iv) `(A xx B) nn (S xx T) = ( A nn S) xx (B nn T),` where S and Tare two sets.
(v) If `A ⊆ B,` then `(A xx C) ⊆ (B xx C)`
(vi) If `A ⊆ B,` then `(A xx B) nn (B xx A)= A^2`
(vii) If `A ⊆ B` and `C ⊆ D,` then `A xx C ⊆ B xx D`


`text(Note :)`

1. `AxxB = BxxA`

2. If A has p elements and B has q elements, then `A x B` has `pq` elements.

3. If `A = phi` and `B = phi,` then `A xx B = phi .`

4. Cartesian product of n sets `A_1, A_2, A_3 , ... , A_n` is the set of all ordered n-tuple

`(a_1 , a_2 ,., a_n), \ \ a_i in A_i , i = 1,2,3, ... , n` and is denoted by

`A_1 xx A_2 xx ... xx A_n .`