NCERT Solutions Class 11 Mathematics Sets Exercise 1.5

Question 1:
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}     and     C = {3, 4, 5, 6}. Find
(i) A′           (ii) B′        (iii) (A ∪ C)′         (iv) (A ∪ B)′           (v) (A′)′              (vi) (B – C)′

Answer:
Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.
(i) A′ = U – A = {5, 6, 7, 8, 9}           
(ii) B′ = U – B = {1, 3, 5, 7, 9}      
(iii) A ∪ C = {1, 2, 3, 4, 5, 6}
Now, (A ∪ C)’ = U – (A ∪ C) = {7, 8, 9}        
 (iv) A ∪ B = {1, 2, 3, 4, 6, 8}
Now, (A ∪ B)′ – U – (A ∪ B) = {5, 7, 9}          
(v) (A′)′ = A = {1, 2, 3, 4}             
(vi) B – C = {2, 8}
Now, (B – C)’ = U – (B – C) = {1, 3, 4, 5, 6, 7, 9}

Question 2:
If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :
(i) A = {a, b, c}               (ii) B = {d, e, f, g}             (iii) C = {a, c, e, g}              (iv) D = {f, g, h, a}

Answer:
Given, U = {a, b, c, d, e, f, g, h}
(i) A = {a, b, c}
A’ = U – A = {d, e, f, g, h}
(ii) B = {d, e, f, g}
B’ = U – B = {a, b, c, h}
(iii) C = {a, c, e, g}
C’ = U – C = {b, d, f, h}
(iv) D = {f, g, h, a}
D’ = U – D = {b, c, d, e}

Question 3:
Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {x : x is an even natural number}                (ii) { x : x is an odd natural number }
(iii) {x : x is a positive multiple of 3}                (iv) { x : x is a prime number }
(v) {x : x is a natural number divisible by 3 and 5}
(vi) { x : x is a perfect square }                          (vii) { x : x is a perfect cube}
(viii) { x : x + 5 = 8 }                                             (ix) { x : 2x + 5 = 9}
(x) { x : x ≥ 7 }                                                       (xi) { x : x ∈ N and 2x + 1 > 10 }

Answer:
Let U = N: Set of natural numbers
(i) {x: x is an even natural number}’ = {x: x is an odd natural number}
(ii) {x: x is an odd natural number}’ ́ = {x: x is an even natural number}
(iii) {x: x is a positive multiple of 3}’ ́= {x: x ∈ N and x is not a multiple of 3}
(iv) {x: x is a prime number}’ ́ ={x: x is a positive composite number and x = 1}
(v) {x: x is a natural number divisible by 3 and 5}’ ́ = {x: x is a natural number that is not divisible
by 3 or 5}
(vi) {x: x is a perfect square}’ ́ = {x: x ∈ N and x is not a perfect square}
(vii) {x: x is a perfect cube}’ ́ = {x: x ∈ N and x is not a perfect cube}
(viii) {x: x + 5 = 8}’ ́ = {x: x ∈ N and x ≠ 3}
(ix) {x: 2x + 5 = 9}’ ́ = {x: x ∈ N and x ≠ 2}
(x) {x: x ≥ 7}’ ́ = {x: x ∈ N and x < 7}
(xi) {x: x N and 2x + 1 > 10}’ ́ = {x: x ∈ N and x ≤ 9/2}

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′

Answer:
Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}
(i) A ∪ B = {2, 3, 4, 5, 6, 7, 8}
    (A ∪ B)′ = = U – (A ∪ B) = {1, 9}
A’ = U – A = {1, 3, 5, 7, 9}
B’ = U – B = {1, 4, 6, 8, 9}
A’ ∩ B′ = {1, 9}
So, (A ∪ B)′ = A′∩ B′ = {1, 9}
(ii) A ∩ B = {2}
     (A ∩ B)’ = U – (A ∩ B) = {1, 3, 4, 5, 6, 7, 8, 9}
A’ = U – A = {1, 3, 5, 7, 9}
B’ = U – B = {1, 4, 6, 8, 9}
A’ U B′ = {1, 3, 5, 7, 9} U {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}
So, (A ∩ B)′ = A′ ∪ B′