VERIFYING THE GIVEN STATEMENT INVOLVING CARTESIAN PRODUCT

Question :

Let A = {x ∈ W | x < 2} , B = {x ∈|1 < x ≤ 4} and C = {3, 5} . Verify that

(i) A × (B U C) = (A × B) U (A × C)

(ii) A × (B n C) = (A × B) n (A × C)

(iii) (A U B) × C = (A × C) U (B × C)

Solution :

A = {x ∈ W | x < 2} , B = {x ∈|1 < x ≤ 4} and C = {3, 5} . 

A = {0, 1} , B = {2, 3, 4} and C = {3, 5} . 

(i) A × (B U C) = (A × B) U (A × C)

L.H.S 

(B U C)  =  {2, 3, 4, 5}

A × (B U C)

= {(0, 2) (0, 3) (0, 4) (0, 5)(1, 2) (1, 3) (1, 4) (1, 5)}  --(1)

R.H.S 

 (A × B) = {(0, 2) (0, 3) (0, 4)(1, 2) (1, 3) (1, 4)}

 (A × C) = {(0, 3)(0, 5) (1, 3) (1, 5)}

(A x B) U (A x C)  = {(0, 2) (0, 3) (0, 4) (0, 5)(1, 2) (1, 3) (1, 4) (1, 5)}  --(2)

(1)  =  (2)

Hence proved

(ii) A × (B n C) = (A × B) n (A × C)

A = {0, 1} , B = {2, 3, 4} and C = {3, 5} . 

L.H.S 

(B n C)  =  {3}

A × (B n C)  =  {(0, 3) (1, 3)}  ---(1)

R.H.S

 (A × B) = {(0, 2) (0, 3) (0, 4)(1, 2) (1, 3) (1, 4)}

 (A × C) = {(0, 3)(0, 5) (1, 3) (1, 5)}

 (A × B) n (A × C)  =  {(0, 3) (1, 3)}  ---(2)

(1)  =  (2)

Hence proved.

(iii) (A U B) × C = (A × C) U (B × C)

A = {0, 1} , B = {2, 3, 4} and C = {3, 5} . 

A U B  =  {0, 1, 2, 3, 4}

(A U B) × C  =  { (0, 3)(0, 5)(1, 3)(1, 5)(2, 3)(2, 5)(3, 3)(3, 5)(4, 3)(4, 5) }  -----(1)

 (A × B) = {(0, 2) (0, 3) (0, 4)(1, 2) (1, 3) (1, 4)}

 (B × C) = {(2, 3) (2, 5) (3, 3)(3, 5) (4, 3) (4, 5)}

 (A × C) U (B × C) =  { (0, 3)(0, 5)(1, 3)(1, 5)(2, 3)(2, 5)(3, 3)(3, 5)(4, 3)(4, 5) }  -----(2)

(1)  =  (2)

Hence proved,

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