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

(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(iv) Verify (ii) using Venn diagram.

(ii) 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 intersection of B and C then again we have to find the intersection between A and the results obtained from the intersection 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}

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

= {-1, 4}……………………. (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 intersection between A, B and A, C then again we will do intersection operation between the results obtained from the intersection of A, B and A, C.

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

= {-1, 4, 6}

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

= {-1, 4}

(A ⋂ B) ⋂ (A ⋂ C) = {-1, 4, 6} ⋂ {-1, 4}

= {-1, 4}…………………….. (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)