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Short Intro

In this post, students can find complete NCERT Solutions for Class 11 Maths Chapter 1 Exercise 1.5 – Sets. This exercise covers complement of sets, properties of sets, Venn diagrams and important set identities with easy step-by-step explanations for CBSE exam preparation.

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Quick Information Box

Particular	Details
Class	11
Subject	Mathematics
Chapter	Sets
Exercise	1.5
Board	NCERT / CBSE
Topics Covered	Complement of Sets & Set Identities

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Concepts Used (Topics Covered)

• Complement of Sets
• Universal Set
• Venn Diagram
• Difference of Sets
• De Morgan’s Laws
• Set Identities
• Union & Intersection Properties

The chapter explains complement of a set and operations related to universal sets.

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Important Formulas

Complement of Set

$A’=U-A$

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De Morgan’s First Law

$(A\cup B)’=A’\cap B’$

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De Morgan’s Second Law

$(A\cap B)’=A’\cup B’$

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Universal Set Property

$A\cup A’=U$

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Null Set Property

$A\cap A’=\phi$

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Questions & Step-by-step Solutions

Question 1

Let:

U = {1,2,3,4,5,6,7,8,9,10}
A = {1,3,5,7,9}

Find A′.

Solution

Complement means elements of U which are not in A.

Therefore,

A′ = {2,4,6,8,10}

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Question 2

Let:

U = {a,b,c,d,e,f,g}
A = {a,c,e,g}

Find A′.

Solution

Elements not present in A are:

A′ = {b,d,f}

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Question 3

If:

U = {1,2,3,4,5,6,7,8}
A = {2,4,6,8}

Find:

(i) A ∪ A′

Solution

A ∪ A′ = U

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(ii) A ∩ A′

Solution

A ∩ A′ = φ

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Question 4

Verify De Morgan’s Laws.

Let:

U = {1,2,3,4,5,6}
A = {1,2,3}
B = {3,4,5}

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(i)

Verify:

$(A\cup B)’=A’\cap B’$

Solution

First,

A ∪ B = {1,2,3,4,5}

Therefore,

(A ∪ B)' = {6}

Now,

A' = {4,5,6}
B' = {1,2,6}

Hence,

A' ∩ B' = {6}

Therefore,

(A ∪ B)' = A' ∩ B'

Verified.

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(ii)

Verify:

$(A\cap B)’=A’\cup B’$

Solution

A ∩ B = {3}

Therefore,

(A ∩ B)' = {1,2,4,5,6}

Now,

A' ∪ B' = {1,2,4,5,6}

Hence,

(A ∩ B)' = A' ∪ B'

Verified.

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Question 5

If:

U = {1,2,3,4,5,6,7,8,9}
A = {2,4,6,8}
B = {1,2,3,4}

Find:

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(i)

A ∪ B

Solution

A ∪ B = {1,2,3,4,6,8}

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(ii)

A ∩ B

Solution

A ∩ B = {2,4}

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(iii)

A - B

Solution

A - B = {6,8}

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(iv)

B - A

Solution

B - A = {1,3}

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Question 6

Show that:

$A\cup \phi=A$

Solution

Union with empty set does not add any element.

Therefore,

A ∪ φ = A

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Question 7

Show that:

$A\cap U=A$

Solution

Every element of A belongs to U.

Hence,

A ∩ U = A

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Common Mistakes

• Forgetting universal set while finding complement
• Confusing complement with difference of sets
• Writing repeated elements
• Incorrect use of Venn diagrams
• Forgetting De Morgan’s laws

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Exam Tips

• Learn all set formulas properly.
• Practice Venn diagram questions regularly.
• Always write distinct elements only once.
• Remember complement is taken with respect to universal set.
• Practice De Morgan’s laws carefully.

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Practice MCQs

MCQ 1

If:

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

Then A′ is:

A. {2,4}
B. {1,3,5}
C. φ
D. {2,3}

Answer

A. {2,4}

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MCQ 2

Which relation is correct?

A. A ∪ φ = φ
B. A ∩ U = φ
C. A ∪ φ = A
D. A ∩ A′ = A

Answer

C. A ∪ φ = A

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MCQ 3

If:

A ∩ B = φ

then sets are called:

A. Equal sets
B. Universal sets
C. Disjoint sets
D. Finite sets

Answer

C. Disjoint sets

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FAQ Section

Q1. What is complement of a set?

Complement of set A contains all elements of universal set which are not in A.

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Q2. What is universal set?

The main set containing all elements under consideration is called universal set.

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Q3. What are disjoint sets?

Sets having no common element are called disjoint sets.

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Q4. What is De Morgan’s Law?

These are important laws relating complement, union and intersection of sets.

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