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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