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Mathematics Grade XI

Grade 11 Mathematics

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Chapter 1: Sets

Class 11 • Mathematics • Chapter 1

Sets

The language of well-defined collections, on which the rest of mathematics is built.

CBSE / NCERT Aligned

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Chapter Roadmap

What a Set Is • Types of Sets • Subsets and Power Set • Venn Diagrams • Operations on Sets • Complement and De Morgan • Key Results • Applications

Why Sets Matter

A set is simply a well-defined collection of objects: the vowels of the alphabet, the natural numbers below 10, or the students in a class. The word ‘well-defined’ is important, because for any object we must be able to say clearly whether it belongs to the collection or not. This single idea is the starting point of modern mathematics.

Sets give us a precise language for grouping objects and describing how groups relate to one another. They are the foundation for relations and functions, for probability, and for almost every later topic. In this chapter you will learn how to write sets, the different types of sets, how one set can sit inside another, and how to combine sets using union, intersection, difference and complement, with Venn diagrams to picture it all.

Key idea

A set is a well-defined collection of distinct objects. Each object is an element. We write a ∈ A to mean ‘a belongs to A’ and a ∉ A to mean ‘a does not belong to A’. The order of elements does not matter, and an element is never listed twice.

Key Terms You Must Know

Term

Meaning

Example

Set

A well-defined collection of distinct objects.

A = {1, 2, 3}

Element

An object that belongs to a set, shown with ∈.

2 ∈ A

Roster form

Listing all elements inside curly brackets.

{a, e, i, o, u}

Set-builder form

Describing elements by a common rule.

{x : x is a vowel}

Empty set

A set with no elements, written ∅ or { }.

{x : x is a month with 32 days}

Subset

A is a subset of B if every element of A is in B.

{1, 2} ⊆ {1, 2, 3}

Universal set

The set U containing all objects under discussion.

U = {1, 2, …, 10}

Core Concepts, Step by Step

1. What a Set Is and How to Write It

A set is a well-defined collection of distinct objects, and each object in it is an element. We can describe a set in two ways. In roster form we simply list the elements inside curly brackets, for example {1, 2, 3, 4, 5}. In set-builder form we state a rule that the elements share, for example {x : x is a natural number and x < 6}. Roster form is handy for small sets; set-builder form is better for large or infinite sets where listing is impractical.

2. Types of Sets

Sets come in several kinds. The empty set, written ∅ or { }, has no elements at all. A finite set has a countable number of elements, while an infinite set goes on without end, such as the set of all natural numbers. Two sets are equal when they have exactly the same elements, regardless of order or repetition, so {1, 2, 3} and {3, 2, 1} are equal. Sets with the same number of elements are called equivalent, which is a weaker idea than equality.

3. Subsets, Power Set and Universal Set

A set A is a subset of B, written A ⊆ B, when every element of A is also in B. The empty set is a subset of every set, and every set is a subset of itself. Intervals such as [2, 5] are subsets of the real numbers. The power set of A is the set of all its subsets; if A has n elements, its power set has 2ⁿ elements. The universal set U is the big set that contains all the objects we are currently discussing.

4. Venn Diagrams

A Venn diagram pictures sets as regions inside a rectangle that stands for the universal set U. Overlapping regions show elements shared between sets, and separate regions show elements that belong to only one set. Venn diagrams turn wordy set problems into clear pictures, which is why they are so useful for solving the survey and counting questions that appear in exams. The figure below shows two sets A and B inside U.

A Venn diagram: two sets A and B inside the universal set U

Venn diagram of sets A and B inside the universal set U

Here A = {1, 2, 3, 4} is the left circle and B = {3, 4, 5, 6} is the right circle. The lens where they overlap holds the shared elements 3 and 4, which is the intersection. Everything inside either circle makes up the union, and anything inside the rectangle but outside both circles lies in neither set.

5. Operations on Sets

Sets are combined using three main operations. The union A ∪ B is the set of all elements in A or B (or both). The intersection A ∩ B is the set of elements in both A and B. The difference A − B is the set of elements in A but not in B. These operations obey friendly laws: they are commutative (A ∪ B = B ∪ A), associative, and distributive, just like addition and multiplication of numbers. The table below summarises them.

The main operations on sets

Operation

Symbol

Meaning

Example (A = {1,2,3}, B = {2,3,4})

Union

A ∪ B

in A or B or both

{1, 2, 3, 4}

Intersection

A ∩ B

in both A and B

{2, 3}

Difference

A − B

in A but not in B

{1}

Difference

B − A

in B but not in A

{4}

The same operations shown as shaded Venn diagrams

Union A ∪ B (all shaded)

Intersection A ∩ B (lens shaded)

Difference A − B (left only)

Complement A′ (outside A)

In each diagram the shaded region is the set named beneath it. The union shades both circles, the intersection shades only the overlapping lens, the difference A − B shades the part of A outside B, and the complement A′ shades everything inside U that lies outside A. Reading a set operation as a shaded region is the quickest way to picture what it means.

6. Complement and De Morgan’s Laws

The complement of a set A, written A′, is the set of all elements of the universal set U that are not in A. Complements interact with union and intersection through De Morgan’s laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′. In words, the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements. These laws are used constantly to simplify expressions and to solve Venn diagram problems.

Key Results & Proofs

Three results are used again and again in this chapter and beyond. Each follows from carefully checking which elements belong to which set.

Result 1: Number of Subsets

Statement. A set with n elements has exactly 2ⁿ subsets.

Proof

Think of building a subset one element at a time.

To build a subset, decide for each element whether to include it or leave it out. – two choices per element

There are 2 choices for the first element, 2 for the second, and so on. – independent choices

number of subsets

=

2 × 2 × … × 2 (n times)

multiply the choices

number of subsets

=

2ⁿ

write as a power

So a set of 3 elements has 2³ = 8 subsets, including the empty set and the set itself.

Result 2: De Morgan’s First Law

Statement. For any two sets, (A ∪ B)′ = A′ ∩ B′.

Proof

Chase a typical element and see which sets it can belong to.

Take any element x in (A ∪ B)′. – start with the left side

Then x is not in A ∪ B, so x is in neither A nor B. – meaning of the complement of a union

So x is in A′ and x is in B′, that is x is in A′ ∩ B′. – x avoids both sets

Every step reverses, so the two sets contain exactly the same elements. – the argument works both ways

(A ∪ B)′

=

A′ ∩ B′

therefore they are equal

The companion law (A ∩ B)′ = A′ ∪ B′ is proved in exactly the same way.

Result 3: The Counting Formula

Statement. For two finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Proof

Think about which elements get counted more than once when you add the sizes.

Adding n(A) and n(B) counts every element of the overlap A ∩ B twice. – the shared part is double counted

To correct this, subtract the overlap once. – remove one extra copy

n(A ∪ B)

=

n(A) + n(B) − n(A ∩ B)

the counting formula

This inclusion and exclusion idea is the key to almost every Venn diagram word problem.

Worked Examples

Example 1

Question: Write the set of vowels in the English alphabet in roster form.

▶ Show full working

List the elements inside curly brackets.

The vowels are a, e, i, o, u. – identify the elements

In roster form: {a, e, i, o, u}. – write inside brackets

Answer: {a, e, i, o, u}.

Example 2

Question: Write {x : x is a natural number and x < 6} in roster form.

▶ Show full working

List the natural numbers that fit the rule.

Natural numbers below 6 are 1, 2, 3, 4, 5. – apply the rule

In roster form: {1, 2, 3, 4, 5}. – list them

Answer: {1, 2, 3, 4, 5}.

Example 3

Question: Write the set {2, 4, 6, 8, 10} in set-builder form.

▶ Show full working

Find the common rule the elements obey.

Each element is an even natural number, the largest being 10. – spot the pattern

Set-builder form: {x : x is an even natural number and x ≤ 10}. – write the rule

Answer: {x : x is an even natural number and x ≤ 10}.

Example 4

Question: Are the sets {1, 2, 3} and {3, 2, 1} equal?

▶ Show full working

Two sets are equal when they have the same elements, regardless of order.

Both sets contain exactly the elements 1, 2 and 3. – compare the elements

Order does not matter, so they are equal. – conclusion

Answer: Yes, they are equal sets.

Example 5

Question: List all the subsets of {a, b}.

▶ Show full working

Include the empty set, the single-element sets and the whole set.

Subsets: { }, {a}, {b}, {a, b}. – list every subset

There are 4 of them, which is 2². – count them

Answer: { }, {a}, {b}, {a, b}; the power set has 4 elements.

Example 6

Question: How many subsets does a set with 5 elements have?

▶ Show full working

Use the result that a set of n elements has 2ⁿ subsets.

number of subsets

=

2⁵

apply the formula with n = 5

number of subsets

=

32

evaluate

Answer: 32 subsets.

Example 7

Question: If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, find A ∪ B.

▶ Show full working

The union collects every element in either set, listing each once.

Combine all elements, without repeating any. – union rule

A ∪ B

=

{1, 2, 3, 4, 5, 6}

the result

Answer: A ∪ B = {1, 2, 3, 4, 5, 6}.

Example 8

Question: For the same A and B, find A ∩ B.

▶ Show full working

The intersection keeps only the elements common to both.

Elements in both A and B: 3 and 4. – find the common elements

A ∩ B

=

{3, 4}

the result

Answer: A ∩ B = {3, 4}.

Example 9

Question: For the same A and B, find A − B and B − A.

▶ Show full working

The difference keeps the elements of one set that are not in the other.

A − B: in A but not in B, so 1 and 2. – remove shared elements from A

B − A: in B but not in A, so 5 and 6. – remove shared elements from B

Answer: A − B = {1, 2} and B − A = {5, 6}.

Example 10

Question: Let U = {1, 2, …, 10} and A = {2, 4, 6, 8, 10}. Find A′.

▶ Show full working

The complement keeps every element of U that is not in A.

Elements of U not in A: 1, 3, 5, 7, 9. – take everything outside A

A′

=

{1, 3, 5, 7, 9}

the result

Answer: A′ = {1, 3, 5, 7, 9}.

Example 11

Question: Verify De Morgan’s law (A ∪ B)′ = A′ ∩ B′ for U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, B = {3, 4}.

▶ Show full working

Work out both sides separately and compare.

A ∪ B = {1, 2, 3, 4}, so (A ∪ B)′ = {5, 6}. – left side

A′ = {4, 5, 6} and B′ = {1, 2, 5, 6}. – find the complements

A′ ∩ B′ = {5, 6}. – right side

Both sides equal {5, 6}. – they match

Answer: Both sides equal {5, 6}, so the law holds.

Example 12

Question: In a class of 40 students, 25 like Maths, 20 like Science and 10 like both. How many like at least one of the two?

▶ Show full working

Use the counting formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

n(A ∪ B)

=

25 + 20 − 10

substitute the numbers

n(A ∪ B)

=

35

simplify

Answer: 35 students like at least one of the two subjects.

Where You Meet This in Real Life

Organising and classifying

Sorting people, products or files into groups, such as students who play cricket or hockey, is exactly the language of sets.

Databases and searches

Searching with ‘and’, ‘or’ and ‘not’ matches intersection, union and complement, so every database query is built on set operations.

Surveys and statistics

Counting how many people read newspaper A or B, or both, uses Venn diagrams and the counting formula every day in market research.

Probability

The events in probability are sets of outcomes, so union, intersection and complement carry straight over into the next stage of your studies.

Logic and computing

Computer logic, search engines and circuit design all rest on the same union, intersection and complement that you learn here.

Practice Sets A to D

Practice Set A – Basics

A1. Write the set of the first five odd natural numbers in roster form.

▶ Reveal full working

List them inside curly brackets.

The first five odd natural numbers are 1, 3, 5, 7, 9.

In roster form: {1, 3, 5, 7, 9}.

Answer: {1, 3, 5, 7, 9}.

A2. Write {a, e, i, o, u} in set-builder form.

▶ Reveal full working

Describe the common rule.

Each element is a vowel of the English alphabet.

Answer: {x : x is a vowel in the English alphabet}.

A3. Is the set {x : x is a month with 32 days} empty?

▶ Reveal full working

Check whether any object satisfies the rule.

No month has 32 days, so the set has no elements.

It is the empty set.

Answer: Yes, it is the empty set.

A4. How many elements does the power set of {1, 2, 3} have?

▶ Reveal full working

Use 2ⁿ with n = 3.

number

=

2³ = 8

apply the formula

Answer: 8.

Practice Set B – Conceptual

B1. What is the difference between equal sets and equivalent sets?

▶ Reveal full working

Compare elements with number of elements.

Equal sets have exactly the same elements.

Equivalent sets only have the same number of elements.

Answer: Equal sets share all elements; equivalent sets only share the same count.

B2. What is the empty set and how is it written?

▶ Reveal full working

Describe a set with nothing in it.

It is the set with no elements at all.

It is written ∅ or { }.

Answer: The set with no elements, written ∅ or { }.

B3. Is every set a subset of itself, and is the empty set a subset of every set?

▶ Reveal full working

Recall the rules for subsets.

Every element of a set is in the set, so it is a subset of itself.

The empty set has no element to break the rule, so it is a subset of every set.

Answer: Yes to both.

B4. What is the universal set?

▶ Reveal full working

Think about the big set in any discussion.

It is the set containing all the objects under consideration.

Every other set in the discussion is a subset of it.

Answer: The set U that contains all objects under discussion.

Practice Set C – Application / Numerical

C1. If A = {1, 2, 3, 4, 5} and B = {2, 4, 6}, find A ∪ B and A ∩ B.

▶ Reveal full working

Union collects everything; intersection keeps the common elements.

A ∪ B

=

{1, 2, 3, 4, 5, 6}

all elements, each once

A ∩ B

=

{2, 4}

elements in both

Answer: A ∪ B = {1, 2, 3, 4, 5, 6}; A ∩ B = {2, 4}.

C2. Let U = {1, 2, …, 10} and A = {1, 3, 5, 7, 9}. Find A′.

▶ Reveal full working

Take every element of U that is not in A.

A′

=

{2, 4, 6, 8, 10}

the even numbers up to 10

Answer: {2, 4, 6, 8, 10}.

C3. If A = {a, b, c, d} and B = {b, d, e}, find A − B and B − A.

▶ Reveal full working

Remove the shared elements from each set.

A − B

=

{a, c}

in A but not B

B − A

=

{e}

in B but not A

Answer: A − B = {a, c}; B − A = {e}.

C4. Find the number of subsets and proper subsets of {p, q, r, s}.

▶ Reveal full working

A proper subset is any subset except the set itself.

subsets

=

2⁴ = 16

apply the formula with n = 4

proper subsets

=

16 − 1 = 15

leave out the set itself

Answer: 16 subsets and 15 proper subsets.

Practice Set D – HOTS / Word Problems

D1. In a survey of 100 people, 60 read newspaper A, 40 read B and 20 read both. How many read at least one, and how many read neither?

▶ Reveal full working

Use the counting formula, then subtract from the total.

n(A ∪ B)

=

60 + 40 − 20

counting formula

n(A ∪ B)

=

80

at least one

neither

=

100 − 80 = 20

subtract from the total

Answer: 80 read at least one; 20 read neither.

D2. Verify De Morgan’s law (A ∩ B)′ = A′ ∪ B′ for U = {1, …, 6}, A = {1, 2, 3, 4}, B = {3, 4, 5}.

▶ Reveal full working

Work out both sides and compare.

A ∩ B = {3, 4}, so (A ∩ B)′ = {1, 2, 5, 6}. – left side

A′ = {5, 6} and B′ = {1, 2, 6}. – find the complements

A′ ∪ B′ = {1, 2, 5, 6}. – right side

Both sides equal {1, 2, 5, 6}. – they match

Answer: Both sides equal {1, 2, 5, 6}, so the law holds.

D3. A set A has 3 elements. How many subsets and how many proper subsets does it have?

▶ Reveal full working

Use 2ⁿ for subsets, then remove the set itself for proper subsets.

subsets

=

2³ = 8

apply the formula

proper subsets

=

8 − 1 = 7

leave out A itself

Answer: 8 subsets and 7 proper subsets.

D4. Show that if A ⊆ B and B ⊆ A, then A = B.

▶ Reveal full working

Use the meaning of a subset in both directions.

A ⊆ B means every element of A is in B.

B ⊆ A means every element of B is in A.

So the two sets have exactly the same elements.

Therefore A = B.

Answer: They contain the same elements, so A = B.

Chapter Summary

Everything in One Glance

What a Set Is

A well-defined collection of distinct objects, written in roster or set-builder form.

Types of Sets

Empty, finite, infinite and equal sets; order and repetition do not matter.

Subsets and Power Set

A ⊆ B if every element of A is in B; a set of n elements has 2ⁿ subsets.

Venn Diagrams

Pictures of sets inside the universal set U, ideal for word problems.

Operations

Union (in either), intersection (in both), difference (in one only), complement (outside).

De Morgan and Counting

(A ∪ B)′ = A′ ∩ B′, (A ∩ B)′ = A′ ∪ B′, and n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Are You Exam-Ready?

8-Point Exam Quick-Check

Write the set of the first four prime numbers in roster form.

Write {1, 4, 9, 16, 25} in set-builder form.

How many subsets does a set with 4 elements have?

If A = {1, 2, 3} and B = {2, 3, 4}, find A ∪ B and A ∩ B.

Find A − B for the sets in the previous question.

With U = {1, …, 8} and A = {2, 4, 6, 8}, find A′.

State both of De Morgan’s laws.

If 30 like tea, 25 like coffee and 10 like both, how many like at least one?

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Class 11 Mathematics Chapter 1: Sets, Complete Notes and Practice

These free Class 11 Maths Chapter 1 Sets notes follow the latest NCERT 2026-27 syllabus and give complete, exam-ready coverage for board exams and competitive foundations. Includes worked examples, step-by-step proofs and graded practice, free on SchoolRevise.com.