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Short Intro

In this post, students can find complete step-by-step solutions for Class 11 Maths Chapter 1 Exercise 1.3 – Sets based on the latest NCERT syllabus. This exercise covers subsets, proper subsets, universal sets, intervals, Venn diagrams, and set relations in a simple and exam-oriented format.

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Quick Information Box

Item	Details
Board	NCERT / CBSE
Class	11
Subject	Mathematics
Chapter	Sets
Exercise	1.3
Main Topics	Subsets & Intervals

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Concepts Used (Topics Covered)

• Subsets
• Proper Subsets
• Universal Set
• Singleton Set
• Empty Set
• Intervals on Real Number Line
• Open & Closed Intervals
• Set Relations
• Venn Diagrams

The exercise mainly focuses on subsets and interval representation.

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Important Formulas

Subset Definition

$A\subset B\iff x\in A\Rightarrow x\in B$

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Open Interval

$(a,b)=\{x:a<x<b\}$

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Closed Interval

$[a,b]=\{x:a\le x\le b\}$

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Complement of Set

$A’=U-A$

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Questions & Step-by-step Solutions

Question 1

Fill in the symbols ⊂ or ⊄.

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(i)

{2,3,4} ⊂ {1,2,3,4,5}

because every element of first set belongs to second set.

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(ii)

{a,b,c} ⊄ {b,c,d}

because a does not belong to second set.

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(iii)

Students of Class XI of your school:

⊂

students of your school.

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(iv)

Circles in a plane:

⊄

circles having radius 1 unit.

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(v)

Triangles in a plane:

⊄

rectangles in the plane.

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(vi)

Equilateral triangles:

⊂

triangles in the plane.

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(vii)

Even natural numbers:

⊂

integers.

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Question 2

State True or False.

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(i)

{a,b} ⊄ {b,c,a}

Solution

Both a and b belong to second set.

False

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(ii)

{a,e} ⊂ vowels

Solution

True

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(iii)

{1,2,3} ⊂ {1,3,5}

Solution

2 does not belong to second set.

False

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(iv)

{a} ⊂ {a,b,c}

Solution

True

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(v)

{a} ∈ {a,b,c}

Solution

a is element, not {a}

False

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(vi)

Even natural numbers less than 6:

⊂

natural numbers dividing 36.

Solution

True

because 2 and 4 divide 36.

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Question 3

Let:

A = {1,2,{3,4},5}

Find incorrect statements.

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(i)

{3,4} ⊂ A

Solution

3 and 4 are not individual elements of A.

Incorrect

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(ii)

{3,4} ∈ A

Solution

Correct

because {3,4} itself is an element of A.

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(iii)

{{3,4}} ⊂ A

Solution

Correct

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(iv)

1 ∈ A

Solution

Correct

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(v)

1 ⊂ A

Solution

1 is not a set.

Incorrect

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(vi)

{1,2,5} ⊂ A

Solution

Correct

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(vii)

{1,2,5} ∈ A

Solution

Not an element of A.

Incorrect

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(viii)

{1,2,3} ⊂ A

Solution

3 is not direct element of A.

Incorrect

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(ix)

φ ∈ A

Solution

Empty set is not element of A.

Incorrect

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(x)

φ ⊂ A

Solution

Empty set is subset of every set.

Correct

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(xi)

{φ} ⊂ A

Solution

φ is not element of A.

Incorrect

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Question 4

Write all subsets.

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(i)

{a}

Solution

φ , {a}

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(ii)

{a,b}

Solution

φ , {a}, {b}, {a,b}

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(iii)

{1,2,3}

Solution

φ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

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(iv)

φ

Solution

φ

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Question 5

Write as intervals.

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(i)

$\{x:x\in R,-4<x\le6\}$

Solution

(−4,6]

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(ii)

$\{x:x\in R,-12<x<-10\}$

Solution

(−12,−10)

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(iii)

$\{x:x\in R,0\le x<7\}$

Solution

[0,7)

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(iv)

$\{x:x\in R,3\le x\le4\}$

Solution

[3,4]

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Question 6

Write intervals in set-builder form.

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(i)

(−3,0)

Solution

$\{x:-3<x<0\}${x:−3<x<0}

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(ii)

[6,12]

Solution

$\{x:6\le x\le12\}$

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(iii)

(6,12]

Solution

$\{x:6<x\le12\}$

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(iv)

[−23,5)

Solution

$\{x:-23\le x<5\}$

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Question 7

Universal set for right triangles and isosceles triangles.

Solution

Universal set can be the set of all triangles.

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Question 8

Select universal sets for A, B and C.

Solution

Valid universal sets:

(i) and (iii)

because they contain all elements of A, B and C.

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Common Mistakes

• Confusing ∈ and ⊂
• Treating element as subset
• Wrong interval notation
• Forgetting empty set is subset of every set
• Writing incorrect subsets

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Exam Tips

• Learn interval symbols carefully.
• Practice subset questions daily.
• Remember:

Element ≠ Subset

• Use Venn diagrams for clarity.

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Practice MCQs

MCQ 1

Which is subset of every set?

A. Universal set
B. Empty set
C. Singleton set
D. Infinite set

Answer:

B. Empty set

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MCQ 2

Which interval represents:

$0\le x<5$

A. (0,5)
B. [0,5]
C. [0,5)
D. (0,5]

Answer:

C. [0,5)

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MCQ 3

If:

A={1,2}

then number of subsets is:

A. 2
B. 3
C. 4
D. 5

Answer:

C. 4

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FAQ Section

What is subset?

If every element of A belongs to B, then A is subset of B.

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What is proper subset?

If:

A ⊂ B and A ≠ B

then A is proper subset of B.

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What is universal set?

The main set under consideration is called universal set.

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What is an interval?

A set of real numbers lying between two numbers.

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