Class 11 Maths Relation Functions Cartesian product of Set

Cartesian product of Set

Cartesian product of sets is the product of two sets. 

Given two non-empty sets P and Q. The cartesian product P × Q is the set of all ordered pairs whose first component is a member of P & second component is a member of Q. i.e., P × Q = { (p,q) : p ∈ P, q ∈ Q }

 If either P or Q is null set, then P × Q will also be empty set, i.e., P × Q = φ

 

Let’s assume 2 sets

  • Man ={Ram, Shyam}
  • Woman {Sita, Gita, Rita}.

 Now Man wants to marry Woman.  The Cartesian product of set Man X Woman is

= {Ajay, Bijay} X  {Carol, Dancy, Ellyn}

= {(Ajay, Carol), (Ajay, Dancy), (Ajay, Ellyn), (Bijay, Carol), (Bijay, Dancy), (Bijay, Ellyn)}

Points to note for Cartesian product

  • Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.
  • A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.
  • A x {infinite set} = {infinite set} where A is non-empty set.
  • n(A x B)  = n(A) * n(B)
  • n(A x B x C)  = n(A) * n(B) * n(C)
  • AxB ≠ BxA

 

Numerical: If (x + 1, y – 2) = (3,1), find the values of x and y.

Solution: x+1 = 3 or x=2

y-2 =1  or y=3

 

Numerical: If X = {x, y, z} and P = {p}, form the sets X × P and P × X.

Solution:

X × P = {  (x,p),  (y,p),   (z,p)  }

P X X =  {  (p,x),   (p,y),   (p,z)  }

Numerical: Let A = {1,2,3}, B = {3,4} and C = {4,5,6}. Find A × (B ∩ C) 

Solution: Let’s first find (B ∩ C)  

(B ∩ C)  = {4}

A × (B ∩ C)   = {1,2,3} X {4}     =  {(1,4),  (2,4),  (3,4)}

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