# Sets Exercise 1.5 Solutions – NCERT Class 11 Mathematics Chapter 1

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1. Let 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}
Find

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

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

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

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

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

(vi) (B – C)′
(B – C)′ = ({2, 4, 6, 8} – {3, 4, 5, 6})’ = {2, 8}’ = {1, 3, 4, 5, 6, 7, 9}
(B – C)′ = {1, 3, 4, 5, 6, 7, 9}`

2. If U = {a, b, c, d, e, f, g, h}
Find the complements of the following sets:

(i) A = {a, b, c}
Complement of A = A′ = {a, b, c}′ = {d, e, f, g, h}

(ii) B = {d, e, f, g}
Complement of B = B′ = {d, e, f, g}′ = {a, b, c, h}

(iii) C = {a, c, e, g}
Complement of C = C′ = {a, c, e, g}′ = {b, d, f, h}

(iv) D = {f, g, h, a}
Complement of D = D′ = {f, g, h, a}′ = {b, c, d, e}

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}
{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 is not a positive multiple of 3}

(iv) {x : x is a prime number}
{x : x is a positive composite number or x = 1}

(v) {x : x is a natural number divisible by 3 and 5}
{x : x is neither a multiple of 3 nor multiple of 5}

(vi) {x : x is a perfect square}
{x : x is a Natural Number and its not a perfect square}

(vii) {x : x is a perfect cube}
{x : x is a Natural Number and its not a perfect cube}

(viii) {x : x + 5 = 8}
{x : x is a Natural Number and x is not equal to 3}

(ix) {x : 2x + 5 = 9}
{x : x is a Natural Number and x is not equal to 2}

(x) {x : x ≥ 7}
{x : x is a Natural Number and it’s less than 7}

(xi) {x : x ∈ N and 2x + 1 > 10}
{x : x is a Natural Number and it’s greater than 4}

4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {2, 4, 6, 8}
B = {2, 3, 5, 7}
Verify that

(i) (A ∪ B)′ = A′ ∩ B′
Let’s solve both sides of this equation and then compare
(A ∪ B)′ = ({2, 4, 6, 8} {2, 3, 5, 7})′ = {2, 3, 4, 5, 6, 7, 8}′ = {1, 9}
A′ ∩ B′ = {2, 4, 6, 8}{2, 3, 5, 7}′ = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} = {1, 9}
Hence (A ∪ B)′ = A′ ∩ B′ = {1, 9}

(ii) (A ∩ B)′ = A′ ∪ B′
(A ∩ B)′ = ({2, 4, 6, 8} {2, 3, 5, 7})′ = {2}′ = {1, 3, 4, 5, 6, 7, 8, 9}
A′ ∪ B′ = {2, 4, 6, 8}{2, 3, 5, 7}′ = {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}
Hence (A ∩ B)′ = A′ ∪ B′ = {1, 3, 4, 5, 6, 7, 8, 9}

5. Draw appropriate Venn diagram for each of the following :

(i) (A ∪ B)′

(ii) A′ ∩ B′

(iii) (A ∩ B)′

(iv) A′ ∪ B′

6. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?
U = {Set of all triangles in a plane}
A = {Set of all triangles in a plane with at least one angle different from 60°}
A′ = U – A = {Set of all Equilateral Triangles}

7. Fill in the blanks to make each of the following a true statement:
(i) A ∪ A′ = . . .
A ∪ A′ = U

(ii) φ′ ∩ A = . . .
φ′ ∩ A = U ∩ A = A

(iii) A ∩ A′ = . . .
A ∩ A′ = φ

(iv) U′ ∩ A = . . .
U′ ∩ A = φ ∩ A = φ