Q. 11 A5.0( 1 Vote )

# For A = {- 3,- 1, 0, 4,6,8,10}, B = {- 1,- 2, 3,4,5,6} and C = {- 1, 2,3,4,5,7}, show that

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(iii) Verify (i) using Venn diagram

Answer :

(i) A ⋃ (B ⋃ C) = (A ⋃ B) ⋃ (A ⋃ C)

__L.H.S.__

The statement A ⋃ (B ⋃ C) can be split into two parts for ease of solving. So using the given data first we will find the union of B and C then again we have to find the union between A and the results obtained from the union of B and C.

Data given

A = {- 3,-1, 0, 4, 6, 8, 10},

B = {- 1,-2, 3, 4, 5, 6}

C = {- 1, 2, 3, 4, 5, 7}

So (B ⋃ C) = {-1,-2, 3, 4, 5, 6} ⋃ {-1, 2, 3, 4, 5, 7}

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

Now A ⋃ (B ⋃ C) = {-3, -1, 0, 4, 6, 8, 10} ⋃ {-1,-2, 3, 4, 5, 6, 7}

= {-3, -2, -1, 0, 3, 4, 5, 6, 7, 8, 10}…….(i)

__R.H.S.__

The statement (A ⋃ B) ⋃ (A ⋃ C) can be split into three parts for ease of solving. First we will find the union between A, B and A, C then again we will do union operation between the results obtained from the union of A, B and A, C

So (A ⋃ B) = {-3, -1, 0, 4, 6, 8, 10} ⋃ {-1, -2, 3, 4, 5, 6}

= {-3, -2, -1, 0, 3, 4, 5, 6, 8, 10}

(A ⋃ C) = {-3, -1, 0, 4, 6, 8, 10} ⋃ {-1, 2, 3, 4, 5, 7}

= {-3, -1, 0, 3, 4, 5, 7, 6, 8, 10}

(A ⋃ B) ⋃ (A ⋃ C) = {-3, -2, -1, 0, 4, 5, 6, 8, 10} ⋃

{-3, -1, 0, 3, 4, 5, 7, 6, 8, 10}

= {-3, -2, -1, 0, 3, 4, 5, 6, 7, 8, 10}….(ii)

From statement (i) and (ii) its verified that L.H.S equals to R.H.S

I.e. A ⋃ (B ⋃ C) = (A ⋃ B) ⋃ (A ⋃ C)

**Venn diagram** for A ⋃ (B ⋃ C) = (A ⋃ B) ⋃ (A ⋃ C)

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For A = {- 3,- 1, 0, 4,6,8,10}, B = {- 1,- 2, 3,4,5,6} and C = {- 1, 2,3,4,5,7}, show that

(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(iii) Verify (i) using Venn diagram

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