PROOFS INVOLVING CARTESIAN PRODUCTS OF SETS

Question :

Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime number. Verify that

(i) (A n B) × C = (A × C) n (B × C)

(ii) A × (B −C) = (A × B) − (A × C)

Solution :

A = The set of all natural numbers less than 8.

Natural number starts with 1.

A = {1, 2, 3, 4, 5, 6, 7}

B = The set of all prime numbers less than 8.

A number which is divisible by 1 and itself are known as prime numbers.

B = {2, 3, 5, 7}

C = The set of even prime number

C = {2}   2 is the one and only even prime number. 

(i) (A n B) × C = (A × C) n (B × C)

L.H.S 

(A n B)  =  {2, 3, 5, 7}

 (A n B) × C = {(2, 2) (3, 2) (5, 2) (7, 2)}  ----(1)

R.H.S

(A × C) = {(1, 2)(2, 2)(3, 2)(4, 2) (5, 2)(6, 2)(7, 2)}

(B × C) = {(2, 2)(3, 2)(5, 2)(7, 2)}

(A × C) n (B × C) = {(2, 2) (3, 2) (5, 2) (7, 2)}  ----(2)

(1)  =  (2)

Hence proved.

(ii) A × (B − C) = (A × B) − (A × C)

A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 3, 5, 7} and C = {2}

Solution :

(B - C) = {3, 5, 7}.

A × (B - C) :

  = {(1, 3) (1, 5) (1, 7) (2, 3) (2, 5) (2, 7) (3, 3) (3, 5) (3, 7) (4, 3) (4, 5) (4, 7) (5, 3) (5, 5) (5, 7) (6, 3) (6, 5) (6, 7)(7, 3) (7, 5) (7, 7)} ----(1)

(A × B) = {(1, 2)(1, 3)(1, 5)(1, 7)(2, 2)(2, 3)(2, 5)(2, 7)(3, 2)(3, 3)(3, 5)(3, 7)(4, 2)(4, 3)(4, 5)(4, 7)(5, 2)(5, 3)(5, 5)(5, 7)(6, 2)(6, 3)(6, 5)(6, 7)(7, 2)(7, 3)(7, 5)(7, 7)

(A × C) = {(1, 2)(2, 2)(3, 2)(4, 2) (5, 2)(6, 2)(7, 2)}

 (A × B) - (A × C)

  =  {(1, 3) (1, 5) (1, 7) (2, 3) (2, 5) (2, 7) (3, 3) (3, 5) (3, 7) (4, 3) (4, 5) (4, 7) (5, 3) (5, 5) (5, 7) (6, 3) (6, 5) (6, 7)(7, 3) (7, 5) (7, 7)} ----(2)

(1)  =  (2)

Hence proved.

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