NCERT Solutions Class 11 Maths Chapter 1 Exercise 1.2

Q1: Which of the following are examples of the null set?

(i) Set of odd natural numbers divisible by 2

Odd numbers = {1, 3, 5, 7, …}
None are divisible by 2.
Result: Null Set

(ii) Set of even prime numbers

Prime numbers = {2, 3, 5, 7, …}
Even primes = {2}.
Since it contains an element ‘2’, it is not empty.
Result: Not a Null Set

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

A number cannot be simultaneously less than 5 AND greater than 7.
No such number exists.
Result: Null Set

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

Parallel lines never intersect, so they have no common point.
Result: Null Set

Q2: Which of the following sets are finite or infinite?

(i) The set of months of a year

Set = {January, February, …, December}.
Contains 12 elements (countable).
Result: Finite

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

This is the set of natural numbers $\mathbb{N}$.
It continues indefinitely.
Result: Infinite

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

Contains elements from 1 to 100.
The number of elements is 100 (countable).
Result: Finite

(iv) The set of positive integers greater than 100

Set = {101, 102, 103, …}.
No upper limit.
Result: Infinite

(v) The set of prime numbers less than 99

Set = {2, 3, 5, …, 97}.
The list is countable and definite.
Result: Finite

Q3: State whether each set is finite or infinite.

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

We can draw infinite lines parallel to the x-axis ($y = k$ where $k \in \mathbb{R}$).
Result: Infinite

(ii) The set of letters in the English alphabet

There are 26 letters (A to Z).
Result: Finite

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

Set = {5, 10, 15, 20, …}.
Result: Infinite

(iv) The set of animals living on the earth

Although the number is extremely large, it is a specific, countable number at any given moment.
Result: Finite

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

Infinite circles can be drawn passing through a single point.
Result: Infinite

Q4: State whether A = B or not.

(i) $A = \{ a, b, c, d \}$, $B = \{ d, c, b, a \}$

The order of elements does not matter in sets.
Elements are exactly the same.
Result: Yes (A = B)

(ii) $A = \{ 4, 8, 12, 16 \}$, $B = \{ 8, 4, 16, 18 \}$

$12 \in A$ but $12 \notin B$.
$18 \in B$ but $18 \notin A$.
Result: No (A $\neq$ B)

(iii) $A = \{2, 4, 6, 8, 10\}$, $B = \{ x : x \text{ is positive even integer and } x \le 10\}$

B in roster form = {2, 4, 6, 8, 10}.
Elements match perfectly.
Result: Yes (A = B)

(iv) $A = \{ x : x \text{ is a multiple of } 10\}$, $B = \{ 10, 15, 20, 25, 30, . . . \}$

$A = \{10, 20, 30, …\}$.
$15 \in B$ but $15 \notin A$.
Result: No (A $\neq$ B)

Q5: Are the following pair of sets equal? Give reasons.

(i) $A = \{2, 3\}$, $B = \{x : x \text{ is solution of } x^2 + 5x + 6 = 0\}$

Solve for B:
$x^2 + 5x + 6 = 0$
$(x + 2)(x + 3) = 0$
$x = -2, -3$
So, $B = \{-2, -3\}$.
$A = \{2, 3\}$.
Since $2 \neq -2$ and $3 \neq -3$.
Result: Not Equal

(ii) $A = \{ x : x \text{ is a letter in the word FOLLOW}\}$, $B = \{ y : y \text{ is a letter in the word WOLF}\}$

$A = \{F, O, L, W\}$ (Repetitions ignored)
$B = \{W, O, L, F\}$
Elements are identical.
Result: Equal

Q6: From the sets given below, select equal sets.

Given Sets:
$A = \{ 2, 4, 8, 12\}$
$B = \{ 1, 2, 3, 4\}$
$C = \{ 4, 8, 12, 14\}$
$D = \{ 3, 1, 4, 2\}$
$E = \{-1, 1\}$
$F = \{ 0, a\}$
$G = \{1, -1\}$
$H = \{ 0, 1\}$

Analysis:
1. Compare B and D:
$B = \{1, 2, 3, 4\}$ and $D = \{1, 2, 3, 4\}$ (Order different, elements same).
$\Rightarrow B = D$

2. Compare E and G:
$E = \{-1, 1\}$ and $G = \{1, -1\}$.
$\Rightarrow E = G$

No other sets are equal.

Answer: B = D and E = G

Sets

1. Identifying Null Sets ($\phi$)

2. Finite vs Infinite Sets

3. Equal Sets ($A = B$)