This document covers the topic of sets in mathematics, introducing fundamental concepts such as types of sets, their representations, and operations on sets like union, intersection, and complement. It emphasizes the significance of sets in various mathematical disciplines and provides practical problems for understanding set operations. Key definitions include empty sets, finite and infinite sets, subsets, and power sets.

1. CLASS- XI
TOPIC: SETS
Prepared By: Mr. Pradeep Sharma, Pgt (Maths)
Kendriya Vidyalaya

2. LEARNING OUTCOMES
• INTRODUCTION
• SETS AND THEIR REPRESENTATIONS
• TYPES OF SETS
• VENN DIAGRAMS
• OPERATIONS ON SETS
• COMPLEMENT OF A SET
• PRACTICAL PROBLEMS

3. Introduction: The theory of sets was developed by German mathematician Georg
Cantor (1845-1918). Sets are used to define the concepts of relations and functions.
The study of geometry, sequences, probability, etc. requires the knowledge of sets.
Sets: A set is a well-defined collection of objects.
For example – i) The vowels in the English alphabet
ii) The solution of the equation: 𝑥2
– 5x + 6 = 0,
Representation of sets: There are two methods to represent a set
1) Roster or tabular form
2) Set-builder form.
Roster or tabular form:- All the elements of a set are listed, the elements are being
separated by commas and are enclosed within braces { }.
Ex: The set of all vowels in the English alphabet is {a, e, i, o, u}.
Set-builder form:- All the elements of a set possess a single common property which is
not possessed by any element outside the set.
Ex:- A = {x : x is a vowel in English alphabet}

4. Types of Sets
The Empty set:- A set which does not contain any element is called the empty
set or the null set or the void set. It is denoted by the symbol ∅ or { }.
Ex:- A = {x : 1 < x < 2, x ∈ 𝑁}
Finite set or Infinite sets:- A set which is empty or consists of a definite number
of elements is called finite otherwise it is infinite.
Ex:- i) W ={x: x is the set of the days of the week} ( finite set)
ii) G ={x: x is the set of points on a line} (infinite set)
All infinite sets cannot be described in the roster form.
Equal Sets:- Two sets A and B are said to be equal if they have exactly the same
elements and we write A = B

5. Subsets
Def: A set A is said to be a subset of a set B if every element of A is also an element of B.
i.e. A ⊂ B if a ∈ A ⇒ a ∈ B
Ex:- The set Q of rational numbers is a subset of the set R of real numbers, therefore Q ⊂ R.
Subsets of set of real numbers
N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R, N ⊄ T.
N : Set of natural numbers
Z: Set of integers
Q: Set of rational numbers
T: Set of irrational numbers
R: Set of real numbers
Power Set:- The collection of all subsets of a set A is called the power set of A. It is denoted
by P(A).
Ex:- A= {1,2} then P(A) = {{ φ,{ 1 }, { 2 }, { 1,2 }}
If A is a set with n(A) = m, then n [ P(A)] = 2 𝑚
.

6. Venn Diagrams
Venn diagrams are named after the English logician, John Venn (1834-1883).
These diagrams consist of rectangles and closed curves usually circles. Most of
the relationships between sets can be represented by means of diagrams which
are known as Venn diagrams.
(VENN DIAGRAM)

7. Operations on Sets
Union of sets:- The union of two sets A and B is the set C which consists of all those
elements which are either in A or in B (including those which are in both).
A ∪ B = { x : x ∈A or x ∈B }
Properties of Union of sets:-
(i) A ∪ B = B ∪ A (Commutative law)
(ii) ( A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law )
(iii) A ∪ φ = A (Law of identity element, φ is the identity of ∪)
(iv) A ∪ A = A (Idempotent law)
(v) U ∪ A = U (Law of U)

8. Operations on Sets
Intersection of sets:- The intersection of two sets A and B is the set of all those elements
which belong to both A and B.
A ∩ B = {x : x ∈ A and x ∈ B}
Properties of Intersection of sets:-
(i) A ∩ B = B ∩ A (Commutative law)
(ii) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law)
(iii) φ ∩ A = φ, U ∩ A = A (Law of φ and U)
(iv) A ∩ A = A (Idempotent law)
(v) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law )

9. Complement of a Set
Let U be the universal set and A a subset of U. Then the complement of A is the set of all
elements of U which are not the elements of A.
A′ = {x : x ∈ U and x ∉ A }
Properties of Complement Sets
1.Complement laws: (i) A ∪ A′ = U (ii) A ∩ A′ = φ
2. De Morgan’s law: (i) (A ∪ B)´ = A′ ∩ B′ (ii) (A ∩ B)′ = A′ ∪ B′
3. Law of double complementation : (A′)′ = A
4. Laws of empty set and universal set φ′ = U and U′ = φ

10. Practical Problems on Union and Intersection of Two Sets
Let A and B be finite sets. If A ∩ B = φ, then
(I) n ( A ∪ B ) = n ( A ) + n ( B )
(ii) n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )
If A, B and C are finite sets, then
n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) +
n ( A ∩ B ∩ C )

11. Some Questions for Practice
1. How many elements has P(A), if A = φ?
2. In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach
mathematics and 4 teach both physics and mathematics. How many teach physics ?
3. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like
tennis only and not cricket? How many like tennis?
4. 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?
5. In a survey of 600 students in a school, 150 students were found to be taking tea and
225 taking coffee, 100 were taking both tea and coffee. Find how many students were
taking neither tea nor coffee?

12. Summary
• A set is a well-defined collection of objects.
• A set which does not contain any element is called empty set.
• A set which consists of a definite number of elements is called finite set,
otherwise, the set is called infinite set.
• Two sets A and B are said to be equal if they have exactly the same elements.
• A set A is said to be subset of a set B, if every element of A is also an element
of B.
• A power set of a set A is collection of all subsets of A. It is denoted by P(A).
• For any two sets A and B, (A ∪ B)′ = A′ ∩ B′ and ( A ∩ B )′ = A′ ∪ B′

13. Thankyou
Prepared By: Mr. Pradeep Sharma, Pgt
(Maths) Kendriya Vidyalaya