If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that
(i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′

Solutions:

The given sets are as follows:

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

(i)  A υ B = {2, 3, 4, 5, 6, 7, 8}

We know that the complement of a set A is denoted by A' and it is equal to U - A,

where U is the universal set.

Thus,

( A υ B)' = U - ( A υ B) =  {1, 9} ..........(1)

A' = U - A = {1, 3, 5, 7, 9}

B' = U - B = {1, 4, 6, 8, 9}

The intersection of two sets is obtained by taking their common elements.

Thus,

A' ∩ B' = {1, 9} ........ (2)

From (1) and (2),

⇒ (A υ B)' = A' ∩ B'

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

A = {2, 4, 6, 8} and B = {2, 3, 5, 7}

A ∩ B = {2}

(A ∩ B)' = U - (A ∩ B) = {1, 3, 4, 5, 6, 7, 8, 9} ....... (3)

We know that the union of two sets is obtained writing all the elements of both sets in a set by removing the duplicates.

A' υ B' = {1, 3, 4, 5, 6, 7, 8, 9} ... (4)

From (3) and (4),

⇒ (A ∩ B)' = A' υ B'

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NCERT Solutions Class 11 Maths Chapter 1 Exercise 1.5 Question 4

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′

Summary:

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}, we have verified that (i) (A ∪ B)′ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′.