Class 11 - Chapter 1: Sets

NCERT Solutions for Class 11 Mathematics Chapter 1 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}

C = {3, 4, 5, 6}

(i) A'

A' = U − A

A' = {5, 6, 7, 8, 9}

(ii) B'

B' = U − B

B' = {1, 3, 5, 7, 9}

(iii) (A ∪ C)'

A ∪ C = {1, 2, 3, 4, 5, 6}

(A ∪ C)' = {7, 8, 9}

(iv) (A ∪ B)'

A ∪ B = {1, 2, 3, 4, 6, 8}

(A ∪ B)' = {5, 7, 9}

(v) (A)'

(A)' = A'

(A)' = {5, 6, 7, 8, 9}

(vi) (B − C)'

B − C = {2, 8}

(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}

A' = {d, e, f, g, h}

(ii) B = {d, e, f, g}

B' = {a, b, c, h}

(iii) C = {a, c, e, g}

C' = {b, d, f, h}

(iv) D = {f, g, h, a}

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}

Complement = {x : x is an odd natural number}

(ii) {x : x is an odd natural number}

Complement = {x : x is an even natural number}

(iii) {x : x is a positive multiple of 3}

Complement = {x : x is a natural number but not a multiple of 3}

(iv) {x : x is a prime number}

Complement = {x : x is a natural number but not a prime number}

(v) {x : x is a natural number divisible by 3 and 5}

Complement = {x : x is a natural number not divisible by both 3 and 5}

(vi) {x : x is a perfect square}

Complement = {x : x is a natural number which is not a perfect square}

(vii) {x : x is a perfect cube}

Complement = {x : x is a natural number which is not a perfect cube}

(viii) {x : x + 5 = 8}

Given x = 3

Set = {3}

Complement = N − {3}

(ix) {x : 2x + 5 = 9}

Given x = 2

Set = {2}

Complement = N − {2}

(x) {x : x ≥ 7}

Complement = {1, 2, 3, 4, 5, 6}

(xi) {x : x ∈ N and 2x + 1 > 10}

2x + 1 > 10

2x > 9

x > 4.5

Set = {5, 6, 7, 8, ...}

Complement = {1, 2, 3, 4}

Question 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'

A ∪ B = {2, 3, 4, 5, 6, 7, 8}

(A ∪ B)' = {1, 9}

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

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

A' ∩ B' = {1, 9}

Hence,

(A ∪ B)' = A' ∩ B'

(ii) (A ∩ B)' = A' ∪ B'

A ∩ B = {2}

(A ∩ B)' = {1, 3, 4, 5, 6, 7, 8, 9}

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

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

A' ∪ B' = {1, 3, 4, 5, 6, 7, 8, 9}

Hence,

(A ∩ B)' = A' ∪ B'

Question 5

Draw appropriate Venn diagrams for the following:

(i) (A ∪ B)'

Shade the region outside both A and B.

(ii) A' ∩ B'

Shade the common region outside both A and B.

(iii) (A ∩ B)'

Shade the entire universal set except the common intersection of A and B.

(iv) A' ∪ B'

Shade every region except A ∩ B.

Question 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'?

Solution:

A contains all triangles that are not equilateral.

Therefore the complement A' consists of triangles whose every angle is 60°.

Hence,

A' = {all equilateral triangles}

Question 7

Fill in the blanks.

(i) A ∪ A' = U

(ii) φ' ∩ A = A

(iii) A ∩ A' = φ

(iv) U' ∩ A = φ