HOW TO CHECK THE GIVEN STATEMENT WITH SUITABLE SETS

Question 1 :

By taking suitable sets A,B,C, verify the following results:

(i) A × (B ∩ C) = (A × B) ∩ (A × C).

Solution :

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

L.H.S

(B n C)  =  {2}

A × (B ∩ C)  =  {1, 2, 3} × {2}

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

R.H.S

(A × B)  = {1, 2, 3} × {1, 2}

(A × B)  =  {(1, 1) (1, 2) (2, 1) (2, 2)(3, 1) (3, 2)}

(A × C)  =  {1, 2, 3} × {2, 3 4}

(A × C) 

  =  {(1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 2) (3, 3) (3, 4)}

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

(1)  =  (2)

Hence proved.  

(ii) A × (B ∪ C) = (A × B) ∪ (A × C)

Solution :

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

L.H.S

(B U C)  =  {1, 2, 3, 4}

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

A × (B ∪ C)  =  {(1, 1) (1, 2) (1, 3) (1, 4)(2, 1) (2, 2) (2, 3) (2, 4)(3, 1) (3, 2) (3, 3) (3, 4)  ------(1)

R.H.S

(A × B)  = {1, 2, 3} × {1, 2}

(A × B)  =  {(1, 1) (1, 2) (2, 1) (2, 2)(3, 1) (3, 2)}

(A × C)  =  {1, 2, 3} × {2, 3 4}

(A × C) 

  =  {(1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 2) (3, 3) (3, 4)}

(A × B) U (A × C)   =  {(1, 1) (1, 2) (1, 3) (1, 4)(2, 1) (2, 2) (2, 3) (2, 4)(3, 1) (3, 2) (3, 3) (3, 4)  ------(2)

(1)  =  (2)

Hence proved.  

(iii) (A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A)

Solution :

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

L.H.S

(A × B)  =  {1, 2, 3} × {1, 2}

=  {(1, 1) (1, 2) (2, 1) (2, 2) (3, 1) (3, 2)}

(B × A)  =  {1, 2} × {1, 2, 3}

=  {(1, 1) (1, 2) (1, 3) (2, 1) (2, 2) (2, 3)}

(A × B) ∩ (B × A)  =  { (1, 1) (1, 2)(2, 1) (2, 2) } -----(1)

R.H.S

(A ∩ B)  =   {1, 2, 3} ∩ {1, 2}  =  {1, 2}

(B ∩ A)  =  {1, 2}

(A ∩ B) × (B ∩ A)  =  { (1, 1) (1, 2)(2, 1) (2, 2) } -----(2)

(1)  =  (2)

Hence proved.

(iv) C − (B − A) = (C ∩ A) ∪ (C ∩ B').

Solution :

Let U = {1, 2, 3, 4, 5}

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

L.H.S

B'  =  {3, 4, 5}

(B - A)  =  { }

 C − (B − A)  =  {2, 3, 4}  ---(1)

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

(C ∩ B')  =  {2, 3, 4}

(C ∩ A) ∪ (C ∩ B')  =  {2, 3, 4}  ---(2)

Hence proved.

(v) (B − A) ∩ C = (B ∩ C) − A = B ∩ (C − A).

Solution :

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

Part 1 :

(B − A)  =  { }

(B − A) ∩ C  =  { }  ------(1)

Part 2 :

(B ∩ C)  =  {2}

(B ∩ C) − A  =  { }   ------(2)

Part 3 :

(C − A)  =  {4}

B ∩ (C − A)  =  { }   ------(3)

(1)  =  (2)  =  (3)

Hence proved.

(vi) (B − A) ∪ C = (B ∪ C) − (A − C)

Solution :

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

L.H.S

(B − A)  =  {3}

(B − A) ∪ C  =  {2, 3, 4}  ------(1)

R.H.S

(B ∪ C)  =  {1, 2, 3, 4}

(A-C)  =  {1}

 (B ∪ C) − (A − C)  =  {2, 3, 4}  ------(2)

(1)  =  (2) 

Hence proved.

Question 2 :

If the universal set U = {1, 3, 5, 7, 9, 11, 13, 15, 17} B = {1, 3, 7, 13, 15} find B'.

Solution :

U = {1, 3, 5, 7, 9, 11, 13, 15, 17}

B = {1, 3, 7, 13, 15}

B' = {5, 9, 11, 17}

Question 3 :

In a survey of 400 students in a school, 100 were listed as taking apple juice. 150 as taking orange juice and 75 were listed as taking both apple as well as orange juice. Find how many students were taking neither apple juice nor orange juice.

Solution :

n(AUB) = 400

n(A) = 100, n(B) = 150, n(AnB) = 75

Number of people who likes apple juice or orange juice

= 100 + 150 - 75

= 250 - 75

= 175

Number of students were taking neither apple juice nor orange juice = 400 - 175

= 225

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