NCERT Solutions for Class 11 Maths Chapter 1 Sets Exercise 1.3

NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.3 Sets is a pivotal part of the curriculum that moves beyond basic identification into the relationship between different sets. This exercise introduces students to the concept of one set being contained within another, which is a fundamental building block for higher mathematics, including Calculus and Probability.

NCERT Solutions for Class 11 Maths Chapter 1 Sets Exercise 1.3

Sets

Easy

Q.

Make correct statements by filling in symbols ⊂ or  in the blank spaces:

$\begin{array}{l}\left(\text{i}\right)\left\{2,3,4\right\}...\left\{1,2,3,4,5\right\}\\ \left(\mathrm{ii}\right)\left\{\text{a},\text{b},\text{c}\right\}...\left\{\text{b},\text{c},\text{d}\right\}\\ \left(\mathrm{iii}\right)\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{student}\mathrm{of}\\ \mathrm{Class}\mathrm{XI}\mathrm{of}\mathrm{your}\mathrm{school}\end{array}\right\}...\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{student}\mathrm{of}\mathrm{your}\\ \mathrm{school}\end{array}\right\}\\ \left(\mathrm{iv}\right)\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{circle}\\ \mathrm{in}\mathrm{the}\mathrm{plane}\end{array}\right\}...\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{circle}\mathrm{in}\mathrm{the}\mathrm{same}\\ \mathrm{plane}\mathrm{with}\mathrm{radius}1\mathrm{unit}\end{array}\right\}\\ \left(\text{v}\right)\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{triangle}\\ \mathrm{in}\mathrm{a}\mathrm{plane}\end{array}\right\}...\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{rectangle}\\ \mathrm{in}\mathrm{the}\mathrm{plane}\end{array}\right\}\\ \left(\mathrm{vi}\right)\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{is}\mathrm{an}\mathrm{equilateral}\\ \mathrm{triangle}\mathrm{in}\mathrm{a}\mathrm{plane}\end{array}\right\}...\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{triangle}\mathrm{in}\\ \mathrm{the}\mathrm{same}\mathrm{plane}\end{array}\right\}\\ \left(\mathrm{vii}\right)\left\{\begin{array}{l}\text{x}:\mathrm{x}\mathrm{is}\mathrm{an}\mathrm{even}\\ \mathrm{natural}\mathrm{number}\end{array}\right\}...\left\{\text{x}:\mathrm{x}\mathrm{is}\mathrm{an}\mathrm{integer}\right\}\end{array}$

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Sets

Easy

Q.

 $\begin{array}{l}{Examine whether the following statements are true or false:}\\ \left({i}\right){ {a,b}}\not\subset {{b,c, a}}\\ \left({ii}\right){ {a,e}}\subset {{x : x is a vowel in the English alphabet}}\\ \left({iii}\right){ }\left\{{ 1, 2, 3 }\right\}\subset {{ 1, 3,5 }}\\ \left({iv}\right){ {a}}\subset {{a,b, c}}\\ \left({v}\right){ {a}}\in {{a,b, c}}\\ \left({vi}\right){ }\left\{\begin{array}{l}{x : x is an even natural }\\ {    number less than 6}\end{array}\right\}\subset \left\{\begin{array}{l}{x : x is a natural number }\\ {     which divides 36}\end{array}\right\}\end{array}$ 

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Sets

Easy

Q.

Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets 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}

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Sets

Easy

Q.

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

(i) A ∪ B
(ii) A ∪ C
(iii) B ∪ C
(iv) B ∪ D
(v) A ∪ B ∪ C
(vi) A ∪ B ∪ D
(vii) B ∪ C ∪ D

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Sets

Difficult

Q.

In a group of students, 100 students know Hindi, 50 know English and 25 know both.
Each of the students knows either Hindi or English. How many students are there in the group?

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Sets

Medium

Q.

In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.

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Sets

Medium

Q.

If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

(i) A ∩ B
(ii) B ∩ C
(iii) A ∩ C ∩ D
(iv) A ∩ C
(v) B ∩ D
(vi) A ∩ (B ∪ C)
(vii) A ∩ D
(viii) A ∩ (B ∪ D)
(ix) (A ∩ B) ∩ (B ∪ C)
(x) (A ∪ D) ∩ (B ∪ C)

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Sets

Easy

Q.

Which of the following pairs of sets are disjoint

(i) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6 }
(ii) {a, e, i, o, u} and { c, d, e, f}
(iii) {x: x is an even integer} and {x: x is an odd integer}

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Sets

Easy

Q.

Show that if A ⊂ B, then C – B ⊂ C – A.

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Sets

Medium

Q.

Let A and B be sets. If A ∩ X = B ∩ X = Φ and A ∪ X = B ∪ X for some set X, show that A = B.

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Designed in accordance with the latest CBSE Class 11 Maths syllabus, Exercise 1.3 focuses on Subsets, Proper Subsets, Intervals as Subsets of R, Power Sets, and the Universal Set. Practicing these problems helps students understand the hierarchy of mathematical structures.

Q1. Make correct statements by filling in the symbols $\subset$ or $\not\subset$ in the blank spaces.

• Definition: A set $A$ is a subset of $B$ if every element of $A$ is also an element of $B$.
• (i) $\{2, 3, 4\} \dots \{1, 2, 3, 4, 5\}$:
  • Answer: $\subset$.
  • Reason: Every element in the first set is also in the second set.
• (ii) $\{a, b, c\} \dots \{b, c, d\}$:
  • Answer: $\not\subset$.
  • Reason: Element '$a$' is not in the second set.
• (iii) $\{x : x \text{ is a student of class XI of your school}\} \dots \{x : x \text{ is a student of your school}\}$:
  • Answer: $\subset$.
  • Reason: Class XI students are inherently part of the school's total student body.
• (iv) $\{x : x \text{ is a circle in the plane}\} \dots \{x : x \text{ is a circle in the same plane with radius 1 unit}\}$:
  • Answer: $\not\subset$.
  • Reason: The first set contains circles of all radii, not just 1 unit.
• (v) $\{x : x \text{ is a triangle in the plane}\} \dots \{x : x \text{ is a rectangle in the plane}\}$:
  • Answer: $\not\subset$.
  • Reason: Triangles cannot be part of a set consisting only of rectangles.
• (vi) $\{x : x \text{ is an equilateral triangle in the plane}\} \dots \{x : x \text{ is a triangle in the same plane}\}$:
  • Answer: $\subset$.
  • Reason: Every equilateral triangle is a triangle.
• (vii) $\{x : x \text{ is an even natural number}\} \dots \{x : x \text{ is an integer}\}$:
  • Answer: $\subset$.
  • Reason: Even natural numbers are always integers.

Q2. Examine whether the following statements are true or false.

• (i) $\{a, b\} \not\subset \{b, c, a\}$: False. Elements '$a$' and '$b$' are in both sets.
• (ii) $\{a, e\} \subset \{x : x \text{ is a vowel in English alphabet}\}$: True. Both are in the set $\{a, e, i, o, u\}$.
• (iii) $\{1, 2, 3\} \subset \{1, 3, 5\}$: False. Element '2' is missing from the second set.
• (iv) $\{a\} \subset \{a, b, c\}$: True. Element '$a$' is in the set.
• (v) $\{a\} \in \{a, b, c\}$: False. $\{a\}$ is a set (subset), not an individual element belonging to the set.
• (vi) $\{x : x \text{ is an even natural number less than 6}\} \subset \{x : x \text{ is a natural number which divides 36}\}$: True. The set $\{2, 4\}$ is a subset of $\{1, 2, 3, 4, 6, 9, 12, 18, 36\}$.

Q3. Let $A = \{1, 2, \{3, 4\}, 5\}$. Which statements are incorrect and why?

• (i) $\{3, 4\} \subset A$: Incorrect. Because $3 \notin A$ and $4 \notin A$; here $\{3, 4\}$ is treated as a single element.
• (ii) $\{3, 4\} \in A$: Correct. $\{3, 4\}$ is listed as an element of $A$.
• (iii) $\{\{3, 4\}\} \subset A$: Correct. The element $\{3, 4\}$ is inside this new set.
• (iv) $1 \in A$: Correct. 1 is an element of $A$.
• (v) $1 \subset A$: Incorrect. An element cannot be a subset; only a set can be a subset.
• (vi) $\{1, 2, 5\} \subset A$: Correct. Every element is in $A$.
• (vii) $\{1, 2, 5\} \in A$: Incorrect. The set $\{1, 2, 5\}$ as a whole is not an element of $A$.
• (viii) $\{1, 2, 3\} \subset A$: Incorrect. 3 is not an element of $A$.
• (ix) $\emptyset \in A$: Incorrect. $\emptyset$ is not an element listed in $A$.
• (x) $\emptyset \subset A$: Correct. $\emptyset$ is a subset of every set.
• (xi) $\{\emptyset\} \subset A$: Incorrect. $\emptyset$ is not an element of $A$.

Q4. Write down all the subsets of the following sets:

• (i) $\{a\}$: $\emptyset, \{a\}$.
• (ii) $\{a, b\}$: $\emptyset, \{a\}, \{b\}, \{a, b\}$.
• (iii) $\{1, 2, 3\}$: $\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{2, 3\}, \{1, 3\}, \{1, 2, 3\}$.
• (iv) $\emptyset$: $\emptyset$.

Q5. How many elements has $P(A)$, if $A = \emptyset$?

• Solution: If $n(A) = 0$, then $n[P(A)] = 2^0 = 1$.
• Answer: $P(A)$ has 1 element.

Q6. Write the following as intervals:

• (i) $\{x : x \in R, -4 < x \le 6\}$: $(-4, 6]$.
• (ii) $\{x : x \in R, -12 < x < -10\}$: $(-12, -10)$.
• (iii) $\{x : x \in R, 0 \le x < 7\}$: $[0, 7)$.
• (iv) $\{x : x \in R, 3 \le x \le 4\}$: $[3, 4]$.

Q7. Write the following intervals in set-builder form.

• (i) $(-3, 0)$: $\{x : x \in R, -3 < x < 0\}$.
• (ii) $[6, 12]$: $\{x : x \in R, 6 \le x \le 12\}$.
• (iii) $(6, 12]$: $\{x : x \in R, 6 < x \le 12\}$.
• (iv) $[-23, 5)$: $\{x : x \in R, -23 \le x < 5\}$.

Q8. What universal set(s) would you propose for:

• (i) The set of right triangles: The set of all triangles or the set of polygons.
• (ii) The set of isosceles triangles: The set of all triangles or the set of two-dimensional figures.

Q9. Given $A = \{1, 3, 5\}, B = \{2, 4, 6\}$ and $C = \{0, 2, 4, 6, 8\}$. Identify the universal set.

• (i) $\{0, 1, 2, 3, 4, 5, 6\}$: Not universal; missing element 8 from $C$.
• (ii) $\emptyset$: Not universal.
• (iii) $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$: Universal set; contains all elements of $A, B$, and $C$.
• (iv) $\{1, 2, 3, 4, 5, 6, 7, 8\}$: Not universal; missing element 0 from $C$.

Key Concepts in Exercise 1.3

To excel in this exercise, students must grasp these specific mathematical definitions:

• Subsets ($\subset$): A set $A$ is a subset of a set $B$ if every element of $A$ is also an element of $B$.

• Power Set $P(A)$: The collection of all subsets of a set $A$. If a set has $n$ elements, its power set has $2^n$ elements.

• Universal Set ($U$): A larger set that contains all the elements under consideration in a particular discussion.

• Intervals: Understanding how to represent subsets of Real Numbers ($R$) using open $(a, b)$ and closed $[a, b]$ brackets.

FAQs – Class 11 Maths Chapter 1 Exercise 1.3 Sets

Q1. What is the difference between $\in$ and $\subset$?

The symbol $\in$ (belongs to) is used for an element of a set, while $\subset$ (is a subset of) is used for a set that is contained within another set.

Q2. How do you calculate the number of subsets for a set with 'n' elements?

The total number of subsets is given by the formula $2^n$. For example, if a set has 3 elements, it will have $2^3 = 8$ subsets.

Q3. What are Intervals in the context of Real Numbers?

Intervals are a way to describe a continuous range of numbers. A closed interval $[a, b]$ includes the endpoints $a$ and $b$, while an open interval $(a, b)$ excludes them.

Q4. Why is the Power Set important?

The Power Set is essential in proving various mathematical theorems and is widely used in combinatorics and the study of sample spaces in probability.