Class 11 - Chapter 1: Sets

NCERT Solutions for Class 11 Mathematics Chapter 1 Sets

Exercise 1.2

Question 1

Which of the following are examples of the null set?

(i) Set of odd natural numbers divisible by 2

Solution:

No odd natural number is divisible by 2.

Therefore, the set contains no element.

Hence, it is a null set.

(ii) Set of even prime numbers

Solution:

2 is an even prime number.

Therefore, the set contains one element:

{2}

Hence, it is not a null set.

(iii) {x : x is a natural number, x < 5 and x > 7}

Solution:

No natural number can be simultaneously less than 5 and greater than 7.

Therefore, the set contains no element.

Hence, it is a null set.

(iv) {y : y is a point common to any two parallel lines}

Solution:

Two parallel lines never intersect.

Therefore, they have no common point.

Hence, it is a null set.

Answer:

The null sets are:

(i), (iii) and (iv)

Question 2

Which of the following sets are finite or infinite?

(i) The set of months of a year

Solution:

There are 12 months in a year.

Hence, it is a finite set.

(ii) {1, 2, 3, ...}

Solution:

The set contains all natural numbers and continues indefinitely.

Hence, it is an infinite set.

(iii) {1, 2, 3, ..., 99, 100}

Solution:

The set contains exactly 100 elements.

Hence, it is a finite set.

(iv) The set of positive integers greater than 100

Solution:

The set is:

{101, 102, 103, ...}

It has infinitely many elements.

Hence, it is an infinite set.

(v) The set of prime numbers less than 99

Solution:

There are only finitely many prime numbers less than 99.

Hence, it is a finite set.

Question 3

State whether each of the following set is finite or infinite.

(i) The set of lines which are parallel to the x-axis

Solution:

Infinitely many lines can be drawn parallel to the x-axis.

Hence, it is an infinite set.

(ii) The set of letters in the English alphabet

Solution:

The English alphabet contains 26 letters.

Hence, it is a finite set.

(iii) The set of numbers which are multiples of 5

Solution:

The multiples of 5 are:

5, 10, 15, 20, ...

The set never ends.

Hence, it is an infinite set.

(iv) The set of animals living on the earth

Solution:

The number of animals living on the earth at a given time is finite.

Hence, it is a finite set.

(v) The set of circles passing through the origin (0, 0)

Solution:

Infinitely many circles can be drawn through the origin.

Hence, it is an infinite set.

Question 4

In the following, state whether A = B or not.

(i)

A = {a, b, c, d}

B = {d, c, b, a}

Solution:

Both sets contain exactly the same elements.

Order of elements does not matter in a set.

Therefore,

A = B

(ii)

A = {4, 8, 12, 16}

B = {8, 4, 16, 18}

Solution:

12 ∈ A but 12 ∉ B

18 ∈ B but 18 ∉ A

Therefore,

A ≠ B

(iii)

A = {2, 4, 6, 8, 10}

B = {x : x is a positive even integer and x ≤ 10}

Solution:

Listing elements of B:

B = {2, 4, 6, 8, 10}

Both sets have exactly the same elements.

Therefore,

A = B

(iv)

A = {x : x is a multiple of 10}

B = {10, 15, 20, 25, 30, ...}

Solution:

A = {10, 20, 30, 40, 50, ...}

B contains 15, 25, 35, ... which are not multiples of 10.

Therefore,

A ≠ B

Final Answers

Question 5

Are the following pairs of sets equal? Give reasons.

(i)

A = {2, 3}

B = {x : x is a solution of x² + 5x + 6 = 0}

Solution:

Given:

x² + 5x + 6 = 0

Factoring,

(x + 2)(x + 3) = 0

Therefore,

x = −2 or x = −3

Hence,

B = {−2, −3}

Since

A = {2, 3}

and

B = {−2, −3}

the elements of the two sets are different.

Therefore,

A ≠ B

(ii)

A = {x : x is a letter in the word FOLLOW}

B = {y : y is a letter in the word WOLF}

Solution:

The distinct letters in the word FOLLOW are:

A = {F, O, L, W}

The distinct letters in the word WOLF are:

B = {W, O, L, F}

Both sets contain exactly the same elements.

Since the order of elements does not matter in a set,

A = B

Question 6

From the sets given below, select equal sets:

A = {2, 4, 8, 12}

B = {1, 2, 3, 4}

C = {4, 8, 12, 14}

D = {3, 1, 4, 2}

E = {−1, 1}

F = {0, a}

G = {1, −1}

H = {0, 1}

Solution:

Two sets are equal if they contain exactly the same elements.

Comparing B and D:

B = {1, 2, 3, 4}

D = {3, 1, 4, 2}

Both contain the same elements.

Therefore,

B = D

Comparing E and G:

E = {−1, 1}

G = {1, −1}

Both contain the same elements.

Therefore,

E = G

No other pair contains exactly the same elements.

Answer:

The equal sets are:

B = D

and

E = G