Q. 14.3( 72 Votes )

Let U = {1,

Answer :


Given that

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}


A = {1, 2, 3, 4}


B = {2, 4, 6, 8}


C = {3, 4, 5, 6}


(i) We have to find complement of A, which is given by (U - A)


A’ = U - A


A’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4}


A’ = {5, 6, 7, 8, 9}


(ii) B’ = U - B


B’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}


B’ = {1, 3, 5, 7, 9}


(iii) (A C)’ = U - (A C)


First we find (A C)


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


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


Now,


(A C)’ = U - (A C)


(A C)’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 5, 6}


(A C)’ = {7, 8, 9}


(iv) (A B)’ = U - (A B)


First we find (A B),


(A B) = {1, 2, 3, 4} {2, 4, 6, 8}


(A B) = {1, 2, 3, 4, 6, 8}


Now,


(A B)’ = U - (A B)


(A B)’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 6, 8}


(A B)’ = {5, 7, 9}


(v)We want to find complement of A’, which is given by U - A’


From part (i), we have A’ = {5, 6, 7, 8, 9}


(A’)’ = U - A’


(A’)’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {5, 6, 7, 8, 9}


(A’)’ = {1, 2, 3, 4} = A


We conclude that (A’)’ = A.


(vi) (B - C)’ = U - (B - C)


First we find (B - C),


(B - C) = {2, 4, 6, 8} - {3, 4, 5, 6}


(B - C) = {2, 8}


Now,


(B - C)’ = U - (B - C)


(B - C)’ = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 8}


(B - C)’ = {1, 3, 4, 5, 6, 7, 9}


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