MATHEMATICS CLASS 11
Class 11 - Chapter 1: Sets
NCERT Solutions for Class 11 Mathematics Chapter 1 Sets
Exercise 1.3
Question 1
Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces.
| Statement | Answer |
|---|---|
| {2, 3, 4} ___ {1, 2, 3, 4, 5} | {2, 3, 4} ⊂ {1, 2, 3, 4, 5} |
| {a, b, c} ___ {b, c, d} | {a, b, c} ⊄ {b, c, d} |
| {x : x is a student of Class XI of your school} ___ {x : x is a student of your school} | ⊂ |
| {x : x is a circle in the plane} ___ {x : x is a circle in the same plane with radius 1 unit} | ⊄ |
| {x : x is a triangle in a plane} ___ {x : x is a rectangle in the plane} | ⊄ |
| {x : x is an equilateral triangle in a plane} ___ {x : x is a triangle in the same plane} | ⊂ |
| {x : x is an even natural number} ___ {x : x is an integer} | ⊂ |
Question 2
Examine whether the following statements are true or false.
(i) {a, b} ⊄ {b, c, a}
Solution:
Since both a and b belong to {b, c, a},
{a, b} ⊂ {b, c, a}
Therefore, the given statement is False.
(ii) {a, e} ⊂ {x : x is a vowel in the English alphabet}
Solution:
The set of vowels is {a, e, i, o, u}.
Since a and e are vowels,
{a, e} ⊂ {a, e, i, o, u}
Therefore, the statement is True.
(iii) {1, 2, 3} ⊂ {1, 3, 5}
Solution:
2 does not belong to {1, 3, 5}.
Therefore, the statement is False.
(iv) {a} ⊂ {a, b, c}
Solution:
The element a belongs to the second set.
Therefore, the statement is True.
(v) a ∈ {a, b, c}
Solution:
a is an element of the set.
Therefore, the statement is True.
(vi) {x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36}
Solution:
Left set = {2, 4}
Divisors of 36 include 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Since 2 and 4 are divisors of 36,
{2, 4} ⊂ {1, 2, 3, 4, 6, 9, 12, 18, 36}
Therefore, the statement is True.
| Part | Answer |
|---|---|
| (i) | False |
| (ii) | True |
| (iii) | False |
| (iv) | True |
| (v) | True |
| (vi) | True |
Question 3
Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?
A = {1, 2, {3, 4}, 5}
Elements of A are:
1, 2, {3,4}, 5
Note that 3 and 4 are not individual elements of A; the set {3,4} is one element of A.
| Statement | Correct / Incorrect | Reason |
|---|---|---|
| (i) {3,4} ⊂ A | Incorrect | 3 and 4 are not elements of A. |
| (ii) {3,4} ∈ A | Correct | {3,4} itself is an element of A. |
| (iii) {{3,4}} ⊂ A | Correct | Its only element {3,4} belongs to A. |
| (iv) 1 ∈ A | Correct | 1 is an element of A. |
| (v) 1 ⊂ A | Incorrect | 1 is a number, not a set. |
| (vi) {1,2,5} ⊂ A | Correct | All elements belong to A. |
| (vii) {1,2,3} ⊂ A | Incorrect | 3 is not an element of A. |
| (viii) φ ⊂ A | Correct | Empty set is a subset of every set. |
| (ix) φ ∈ A | Incorrect | Empty set is not an element of A. |
| (x) φ ⊂ A | Correct | Empty set is a subset of every set. |
| (xi) {φ} ⊂ A | Incorrect | φ is not an element of A. |
Incorrect Statements:
(i), (v), (vii), (ix) and (xi)
Question 4
Write down all the subsets of the following sets.
(i) {a}
Subsets are:
φ, {a}
(ii) {a, b}
Subsets are:
φ, {a}, {b}, {a, b}
(iii) {1, 2, 3}
Subsets are:
φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
(iv) φ
The empty set has only one subset, namely itself.
Subsets:
φ
Question 5
Write the following as intervals.
(i) {x : x ∈ R, −4 < x ≤ 6}
Answer: (−4, 6]
(ii) {x : x ∈ R, −12 < x < −10}
Answer: (−12, −10)
(iii) {x : x ∈ R, 0 ≤ x < 7}
Answer: [0, 7)
(iv) {x : x ∈ R, 3 ≤ x ≤ 4}
Answer: [3, 4]
Question 6
Write the following intervals in set-builder form.
(i) (−3, 0)
{x : x ∈ R, −3 < x < 0}
(ii) [6, 12]
{x : x ∈ R, 6 ≤ x ≤ 12}
(iii) (6, 12]
{x : x ∈ R, 6 < x ≤ 12}
(iv) [−23, 5)
{x : x ∈ R, −23 ≤ x < 5}
Question 7
What universal set(s) would you propose for each of the following?
(i) The set of right triangles
A suitable universal set is the set of all triangles.
(ii) The set of isosceles triangles
A suitable universal set is the set of all triangles.
Question 8
Given:
A = {1, 3, 5}
B = {2, 4, 6}
C = {0, 2, 4, 6, 8}
Which of the following may be considered as universal set for A, B and C?
(i) {0,1,2,3,4,5,6}
(ii) φ
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}
Solution:
A universal set must contain all elements of A, B and C.
A ∪ B ∪ C = {0,1,2,3,4,5,6,8}
Checking options:
(i) Does not contain 8.
(ii) Contains no elements.
(iii) Contains all required elements.
(iv) Does not contain 0.
Therefore, only option (iii) can be taken as a universal set.
Answer:
(iii) {0,1,2,3,4,5,6,7,8,9,10}