PROVE THAT PROBLEMS INVOLVING CARTESIAN PRODUCT

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.

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