PROVE THAT PROBLEMS INVOLVING CARTESIAN PRODUCT

About "Prove that Problems Involving Cartesian Product"

Prove that Problems Involving Cartesian Product :

Here we are going to see, how to prove the given Cartesian product statements. 

Cartesian Product of Two Sets

If A and B are two non-empty sets, then the set of all ordered pairs (a, b) such that a ∈ A, b ∈ B is called the Cartesian Product of A and B, and is denoted by A x B .

Thus, A x B = { (a,b) |a ∈ A,b ∈ B } 

Question 1 :

If A = {5, 6} , B = {4, 5, 6} , C = {5, 6, 7} , Show that A × A = (B × B) n (C × C) .

Solution :

A = {5, 6} , B = {4, 5, 6} , C = {5, 6, 7}

L.H.S 

A = {5, 6} and A = {5, 6}

A x A   =  {(5, 5) (5, 6) (6, 5) (6, 6)  ----(1)

R.H.S 

B = {4, 5, 6} and B = {4, 5, 6}

B × B 

  =  {(4, 4) (4, 5) (4, 6)(5, 4) (5, 5) (5, 6)(6, 4) (6, 5) (6, 6)}

 C = {5, 6, 7} and C = {5, 6, 7}

C × C

  =  {(5, 5)(5, 6)(5, 7)(6, 5)(6, 6)(6, 7)(7, 5)(7, 6)(7, 7)}

(B × B) n (C × C)  =  {(5, 5)(5, 6)(6, 5)(6, 6)}  -----(2)

(1)  =  (2)

L.H.S  =  R.H.S

Question 2 :

Given A = {1, 2, 3}, B = {2, 3, 5}, C = {3, 4} and D = {1, 3, 5}, check if (A n C) × (B n D) = (A × B)n(C × D) is true?

Solution :

In order to check if the given statement is true, let us find values of L.H.S and R.H.S

A = {1, 2, 3}, B = {2, 3, 5}, C = {3, 4} and D = {1, 3, 5}

A n C  means common elements of sets A and C.

A n C  =  {3}

B n D means common elements of sets B and D.

B n D  =  {3, 5}

L. H.S

(A n C) × (B n D)  =  { (3, 3)(3, 5) }  -----(1)

(A × B)

A = {1, 2, 3}, B = {2, 3, 5}

A x B= {(1, 2)(1, 3)(1, 5)(2, 2)(2, 3)(2, 5) (3, 2)(3, 3)(3, 5)}  

(C × D)

C = {3, 4} and D = {1, 3, 5}

C x D= {(3, 1)(3, 3)(3, 5)(4, 1)(4, 3)(4, 5)}

R.H.S 

(A × B)n(C ×D)  =  {(3, 3)(3,5)}  ------(2)

(1)  =  (2)

Hence the given statement is true.

After having gone through the stuff given above, we hope that the students would have understood, "Prove that Problems Involving Cartesian Product".

Apart from the stuff given in this section "Prove that Problems Involving Cartesian Product", if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.