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Class – XI, CBSE MB1101: Sets
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What is a Set?
Class – XI MB1101: Sets
Important Definitions
Topic: Sets

What is a Set?
Class - XI MB1101: Sets
Topic: Sets
A set is a "well defined collection of
objects, where the order in which the
elements are listed is immaterial and
while listing the elements repetition
is not allowed.

What are the
symbols for some
important sets?
Topic: Sets

What are the
symbols for some
important sets?
Topic: Sets
N: the set of all natural numbers
Z: the set of all integers
Q: the set of all rational numbers
R: the set of real numbers
Z+ : the set of positive integers
Q+ : the set of positive rational numbers
R+ : the set of positive real numbers

What is a
Notation?
Topic: Sets

What is a
Notation?
Topic: Sets
If is an element of a set , we say that
" belongs to " the Greek symbol
(epsilon) is used to denote the phrase
' '. Thus, we write . .
If ' ' is not an element of a set , we
write
a A
a A
belongs to a b
b A
b A and read " does not
belong to ".
Thus, in the set = 1, 2, 3 , 1 but
4 .
b
A
A A
A

How to represent a
set in Roster or
Tabular Form?
Topic: Representations of Sets

How to represent a
set in Roster or
Tabular Form?
In roster form, all the elements of a set are
listed, separated by commas and are enclosed
within braces { }. For example, the set of all
even positive integers less than 7 is described
in roster form as {2, 4, 6}. The set of odd
natural numbers is represented by {1, 3, 5, . . .}.
The dots tell us that the list of odd numbers
continue indefinitely.

How to represent a
set in Set-Builder
Form?

How to represent a
set in Set-Builder
Form?
In set-builder form, all the elements of a
set possess a single common property
which is not possessed by any element
outside the set. For example, in the set
{ , , , , }, all the elements possessa e i o u a
common property, namely, each of them is
a vowel in the English alphabet, and no
other letter possess this property. Denoting
this set by , we write
= { : is a vowel in English alphabet}.
V
V x x

What is Empty set?
Topic: Empty Set

What is Empty set?
Topic: Empty Set
We define empty set also called null
or void set, as a set which does not
contain any element. We denote empty
set by the symbol .

What are Finite
and Infinite sets?
Topic: Finite and Infinite Sets

What are Finite
and Infinite sets?
Topic: Finite and Infinite Sets
A set which is either empty or contains
a definite number of element, is called
a finite set. A set which is not finite is
called is an infinite set e.g. the set of
all natural numbers is an infinite set and
1,2,3 is a finite set .A

What are equal
sets?
Topic: Equal Sets

What are equal
sets?
Topic: Equal Sets
Given two sets and , if every element
of is also an element of and if every
element of is also an element of , then
the sets and are said to be equal. In
this case we write = . If two
A B
A B
B A
A B
A B sets are
equal then they have exactly the same
elements. If two sets are not equal then
they are called as unequal sets and we
write A B

What are Subsets?
Topic: Subsets

What are Subsets?
Topic: Subsets
A set is said to be a subset of a set if every
element of is also an element of .
In other words, if whenever , then
. We use the symbol " " which means
implies. Using this symbol, we
A B
A B
A B a A
a B
can write the
definition of as follows:
if .
We read the above statement as " is a subset
of if is an element of implies that is also
an element of ". If is not a subset of
subset
A B a A a B
A
B A A
B A , we
write .
B
A B

What are Subsets?
Topic: Subsets
We use the symbol " " for two way
implications, and read it as if and only also
written as "iff").
So and
It follows from the above definition that
every set , i.e.,
A B A B B A
A is a subset of itself .
Since the empty set has no elements, we
agree to say that is a subset of every set.
We now consider some examples :
Some of the obvious relations among the
subsets of real numbers are
A A
N Z Q R

What is Super Set?
Topic: Subsets

What is Super Set?
Topic: Subsets
Let and be two sets. If and
, then is called a proper subset
of and is called a superset of A.
For example, = {1, 2, 3} is a proper
subset of = {1, 2, 3, 4}.
A B A B
A B A
B B
A
B

What is Singleton
Set?
Topic: Subsets

What is Singleton
Set?
Topic: Subsets
If a set has only one element, we
call it a . Thus,{ a } is a
singleton set.
A
singleton set

Explain Intervals
as a subset of R?
Topic: Subsets

Describe Intervals
as a subset of R?
Topic: Subsets
Let , R and . Then the set of real numbers
: is called an open interval and is
denoted by ( , ). All the points between and
belong to the open interval ( , ) but , themselves
do n
a b a b
y a y b
a b a b
a b a b
ot belong to this interval. The interval which
contains the end points also is called
and is denoted by [ , ]. Thus, , : .
We can also have intervals closed at one end and open
closed interval
a b a b x a x b
at the other, i.e., , : is an
from to , including but excluding .
, : is an open interval from to
including but excluding .
a b x a x b open
interval a b a b
a b x a x b a b
b a

Describe Intervals
as a subset of R?
Topic: Subsets
These notations provide an alternative way of
designating the subsets of set of real numbers.
For example, if 3, 5 and 7, 9 ,
then . The set 0, defines the set of
non-negative real numbers,
A B
A B
while set ,0
defines the set of negative real numbers. The set
, describes the set of real numbers as a line
extending from to . The number ( ) is
called the length of any of the intervals
b a
( , ),
[ , ], [ , ) or ( , ].
a b
a b a b a b

Describe Intervals
as a subset of R?
Topic: Subsets
On real number line, various types of intervals described
above as subsets of , are shown below:
Graphically, an open interval 3,1 is shown as:
R
The interval which contains the end points also is
called closed interval and is denoted by ,
Thus, , : . Graphically, a closed
interval 3,1 is shown as:
a b
a b x a x b

Describe Intervals
as a subset of R?
Topic: Subsets
We can also have closed at one end and
open at the other, i.e., , : is an half-open
interval from to , including but excluding .
, : is an half-open interval f
half - open intervals
a b x a x b
a b a b
a b x a x b rom to
including but excluding . So, the half open interval 2,4
is shown graphically as:
a b
b a

What is Power
Set?
Topic: Power Set

What is Power
Set?
Topic: Power Set
The collection of all subsets of a set is
called the of . It is denoted by
( ). In ( ), every element is a set.
Thus, as in above, if A = { 1, 2 }, then
( ) = { , { 1 }, { 2 }, { 1,2 }}
A
power set A
P A P A
P A
2
.
Also, note that the number of elements
in ( ) = [ ( )] = 4 = 2 .
In general, if is a set with ( ) = , then
it can be shown that [ ( )] = 2 .m
P A n P A
A n A m
n P A

What is Universal
Set?
Topic: Universal Set

What is Universal
Set?
Topic: Universal Set
A universal set is the collection of all
objects in a particular context. All
other sets in that framework constitute
subsets of the universal set, which is
denoted as an uppercase italic letter .
Th
U
e objects themselves are known as
elements or members of .U

What are Venn
Diagrams?
Topic: Venn Diagrams

What are Venn
Diagrams?
Most of the relationships between sets can be represented by
means of diagrams which are known as .
In Venn diagrams, the elements of the sets are written in their
respective circles.
The
Venn diagrams
Venn diagram with three overlapping circle makes 7
different regions as shown below:

What are Venn
Diagrams?
Let us try to understand what these regions indicate:
Region 1 contains elements that are in but not in or .
Thus 1, if , and .
Region 2 contains elements that are in but not in
A B C
x x A x B x C
B A or .
Thus 2, if , and .
Region 3 contains elements that are and but not in .
Thus 3, if , and .
Region 4 contains elements that are in but not in or .
Thus 4, i
C
x x B x A x C
A B C
x x A x B x C
C A B
x f , and .
Thus 5, if , and .
Thus 6, if , and .
R
x C x A x B
A C B
x x A x C x B
B C A
x x B x C x A
egion 7 contains elements that are in , and .
Thus 7, if , and .
A B C
x x A x B x C

What is Union of
Sets?
Topic: Operations on Sets

What is Union of
Sets?
Let and be any two sets. The union of and
is the set which consists of all the elements of
and all the elements of , the common elements
being taken only once. The symbol ' ' is used to
denot
A B A
B
A B
e the . Symbolically, we write
and usually read as ' union '.
Thus, we can define the union of two sets and
as the set C which consists of all those elements
which are either in A or in
union A B
A B
A B
B (including those
which are in both). In symbols, we write
: or .A B x x A x B

What is Union of
Sets?
The union of two sets can be represented
by a Venn diagram as shown in the figure:
The shaded portion in Figure represents .A B

What is
Intersection of
Sets?

What is
Intersection of
Sets?
The intersection of sets and is the set of all
elements which are common to both and .
The symbol ' ' is used to denote the .
The intersection of two sets and is the set of
all t
A B
A B
intersection
A B
hose elements which belong to both and .
Symbolically, we write : and .
Thus, we can define the intersection of two sets
and as the set of all those elements which belong
both A and B.
A B
A B x x A x B
A
B

What is
Intersection of
Sets?
The shaded portion in Figure indicates
the intersection of A and B.

What is Difference
of Sets?

What is Difference
of Sets?
The difference of the sets and in this order is the set of
elements which belong to but not to . Symbolically, we
write and read as" minus ".
Using the set builder notation, we can rewrit
A B
A B
A B A B
e the definition
of difference as : and .
The difference of two sets and can be represented by Venn
diagram as shown in the figure.
A B x x A x B
A B
The shaded portion represents the difference of the two sets and .A B

What is Difference
of Sets?
The sets , and are mutually
disjoint sets, i.e., the intersection of any of these
two sets is the null set as shown in the figure
A B A B B A

What is
Complement of a
Sets?

What is
Complement of a
Sets?
Let be the universal set and is a subset
of . Then the complement of is the set
of all elements of which are not the
elements of A. Symbolically, we write '
to denote the complement of wit
U A
U A
U
A
A h respect
to . Thus, : and .
Obviously . So the complement
of a set can also be seen as difference
between a universal set and the set A.
U A x x U x A
A U A
A

Fact # 1
Two sets A and B are said to be equal if they have exactly the
same elements.
Important Facts
Topic: Equal Sets

Fact # 2
A set A is said to be subset of a set B, if every element of A is also
an element of B.
Important Facts
Topic: Subsets

Fact # 3
Intervals are subsets of R.
Important Facts
Topic: Subsets

Fact # 4
A power set of a set A is collection of all subsets of A. It is denoted
by P(A).
Important Facts
Topic: Power Sets

Fact # 5
The union of two sets A and B is the set of all those elements
which are either in A or in B.
Important Facts

Fact # 6
The intersection of two sets A and B is the set of all elements
which are common. The difference of two sets A and B in this
order is the set of elements which belong to A but not to B.
Important Facts

Fact # 7
The complement of a subset A of universal set U is the set of all
elements of U which are not the elements of A.
Important Facts
Topic: Complement of a Set

Important Formulae
1. Complement law:
( )
( )
i A A U
ii A A

Important Formulae
2. De Morgan's law:
( )
( )
i A B A B
ii A B A B

Important Formulae
3. Law of double complementation:
A A

Important Formulae
4. Laws of empty set and universal set
and .U U

Important Formulae
5. Commutative law
,A B B A A B B A

Important Formulae
6. Law of identity element
, ,A A A U A A

Important Formulae
7. Idempotent law
,A A A A A A

Important Formulae
8.
U A U

Important Formulae
9. Associative Laws:
,A B C A B C A B C A B

Important Formulae
10. Distributive Laws:
,A B C A B A C A B C A B A C

Important Formulae
11. Let and be finite sets. If thenA B A B
n A B n A n B

Important Formulae
12. In general, if and are finite sets, thenA B
.n A B n A n B n A B

Important Formulae
13. For any three finite sets , ,A B C
.
n A B C n A n B n C n A B n B C
n A C n A B C

How to change the representation of the set from Set-
Builder to Roster Form?
Class – XI,CBSE
Important Procedures
MB1101: Sets

Class – XI,CBSE
:Understand the description of the elements of the set.Step1
MB1101: Sets

Class – XI,CBSE
MB1101: Sets
:Identify each member of the set one at a time.Step2

Class – XI,CBSE
MB1101: Sets
: If you can see a pattern after finding a few members, use an ellipsis to write the set in
Roster form so that everyone can see the pattern and guess ALL elements.
Step3

Class – XI,CBSE
MB1101: Sets
: If you can see a pattern after finding a few members, use an ellipsis to write the set in
Roster form so that everyone can see the pattern and guess ALL elements.
Step3
: If there are just a few elements (usually less than 10) then you need to mention all
elements in the Roster form.
Step4

How to change the representation of the set from Roster to
Set-Builder Form?
Class – XI,CBSE
MB1101: Sets

Set-Builder Form?
Class – XI,CBSE
MB1101: Sets
:Find a pattern among all members of the set.Step1

Set-Builder Form?
Class – XI,CBSE
MB1101: Sets
:Find out what are the common characteristics of the members.Step2

Set-Builder Form?
Class – XI,CBSE
MB1101: Sets
:Find out what are the common characteristics of the members.Step2
: Use a group of statements to describe these common characteristics in set-builder form.
This conversion is not always easy because the description of the elements of the set may not be
unique.
Step3

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