Page 1
SETS
2.2 Set-Builder Form
In this form, we write a variable (say x) representing any
member of the set followed by a property satisfied by each
member of the set.
For example, the set A of all prime numbers less than 10 in
the set-builder form is written as
A = {x | x is a prime number less that 10}
The symbol '|' stands for the words 'such that'. Sometimes,
we use the symbol ':' in place of the symbol '|'.
3. TYPES OF SETS
3.1 Empty Set or Null Set
A set which has no element is called the null set or empty
set. It is denoted by the symbol .
For example, each of the following is a null set :
(a) The set of all real numbers whose square is –1.
(b) The set of all rational numbers whose square is 2.
(c) The set of all those integers that are both even and odd.
A set consisting of atleast one element is called a
non-empty set.
3.2 Singleton Set
A set having only one element is called singleton set.
For example, {0} is a singleton set, whose only member is 0.
3.3 Finite and Infinite Set
A set which has finite number of elements is called a finite
set. Otherwise, it is called an infinite set.
For example, the set of all days in a week is a finite set
whereas the set of all integers, denoted by
{............ -2, -1, 0, 1, 2,...} or {x | x is an integer}, is an infinite set.
An empty set which has no element in a finite set A is
called empty of void or null set.
1. SET
A set is a collection of well-defined and well distinguished
objects of our perception or thought.
1.1 Notations
The sets are usually denoted by capital letters A, B, C, etc.
and the members or elements of the set are denoted by lower-
case letters a, b, c, etc. If x is a member of the set A, we write
x
A (read as 'x belongs to A') and if x is not a member of the
set A, we write x A (read as 'x does not belong to A,). If x
and y both belong to A, we write x, y
A.
2. REPRESENTATION OF A SET
Usually, sets are represented in the following two ways :
(i) Roster form or Tabular form
(ii) Set Builder form or Rule Method
2.1 Roster Form
In this form, we list all the member of the set within braces
(curly brackets) and separate these by commas. For example,
the set A of all odd natural numbers less that 10 in the Roster
form is written as :
A = {1, 3, 5, 7, 9}
(i) In roster form, every element of the set is listed
only once.
(ii) The order in which the elements are listed is
immaterial.
For example, each of the following sets denotes
the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}
SETS, RELATIONS & FUNCTIONS
SETS, RELATIONS & FUNCTIONS
Page 2
SETS
2.2 Set-Builder Form
In this form, we write a variable (say x) representing any
member of the set followed by a property satisfied by each
member of the set.
For example, the set A of all prime numbers less than 10 in
the set-builder form is written as
A = {x | x is a prime number less that 10}
The symbol '|' stands for the words 'such that'. Sometimes,
we use the symbol ':' in place of the symbol '|'.
3. TYPES OF SETS
3.1 Empty Set or Null Set
A set which has no element is called the null set or empty
set. It is denoted by the symbol .
For example, each of the following is a null set :
(a) The set of all real numbers whose square is –1.
(b) The set of all rational numbers whose square is 2.
(c) The set of all those integers that are both even and odd.
A set consisting of atleast one element is called a
non-empty set.
3.2 Singleton Set
A set having only one element is called singleton set.
For example, {0} is a singleton set, whose only member is 0.
3.3 Finite and Infinite Set
A set which has finite number of elements is called a finite
set. Otherwise, it is called an infinite set.
For example, the set of all days in a week is a finite set
whereas the set of all integers, denoted by
{............ -2, -1, 0, 1, 2,...} or {x | x is an integer}, is an infinite set.
An empty set which has no element in a finite set A is
called empty of void or null set.
1. SET
A set is a collection of well-defined and well distinguished
objects of our perception or thought.
1.1 Notations
The sets are usually denoted by capital letters A, B, C, etc.
and the members or elements of the set are denoted by lower-
case letters a, b, c, etc. If x is a member of the set A, we write
x
A (read as 'x belongs to A') and if x is not a member of the
set A, we write x A (read as 'x does not belong to A,). If x
and y both belong to A, we write x, y
A.
2. REPRESENTATION OF A SET
Usually, sets are represented in the following two ways :
(i) Roster form or Tabular form
(ii) Set Builder form or Rule Method
2.1 Roster Form
In this form, we list all the member of the set within braces
(curly brackets) and separate these by commas. For example,
the set A of all odd natural numbers less that 10 in the Roster
form is written as :
A = {1, 3, 5, 7, 9}
(i) In roster form, every element of the set is listed
only once.
(ii) The order in which the elements are listed is
immaterial.
For example, each of the following sets denotes
the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}
SETS, RELATIONS & FUNCTIONS
SETS, RELATIONS & FUNCTIONS SETS, RELATIONS & FUNCTIONS
3.4 Cardinal Number
The number of elements in finite set is represented by n(A),
known as Cardinal number.
3.5 Equal Sets
Two sets A and B are said to be equals, written as A = B, if
every element of A is in B and every element of B is in A.
3.6 Equivalent Sets
Two finite sets A and B are said to be equivalent, if n
(A) = n (B). Clearly, equal sets are equivalent but equivalent
sets need not be equal.
For example, the sets A = { 4, 5, 3, 2} and B = {1, 6, 8, 9} are
equivalent but are not equal.
3.7 Subset
Let A and B be two sets. If every elements of A is an element
of B, then A is called a subset of B and we write A B or
B A (read as 'A is contained in B' or B contains A'). B is
called superset of A.
(i) Every set is a subset and a superset itself.
(ii) If A is not a subset of B, we write A B.
(iii) The empty set is the subset of every set.
(iv) If A is a set with n(A) = m, then the number of
subsets of A are 2
m
and the number of proper
subsets of A are 2
m
-1.
For example, let A = {3, 4}, then the subsets of A
are , {3}, {4}. {3, 4}. Here, n(A) = 2 and number
of subsets of A = 2
2
= 4. Also, {3}
{3,4}and {2,3}
{3, 4}
3.8 Power Set
The set of all subsets of a given set A is called the power set
of A and is denoted by P(A).
For example, if A = {1, 2, 3}, then
P(A) = { , {1}, {2}, {3}, {1,2} {1, 3}, {2, 3}, {1, 2, 3}}
Clearly, if A has n elements, then its power set P (A) contains
exactly 2
n
elements.
4. OPERATIONS ON SETS
4.1 Union of Two Sets
The union of two sets A and B, written as A B (read as 'A
union B'), is the set consisting of all the elements which are
either in A or in B or in both Thus,
A B = {x : x
A or x
B}
Clearly, x
A B x
A A or x
B, and
x A B x A A and x B.
For example, if A = {a, b, c, d} and B = {c, d, e, f}, then
A B = {a, b, c, d, e, f}
4.2 Intersection of Two sets
The intersection of two sets A and B, written as A
B
(read as ‘A ’ intersection ‘B’) is the set consisting of all the
common elements of A and B. Thus,
A
B = {x : x
A and x
B}
Clearly, x
A
B x
A A and x
B, and
x
A
B x
A A or x
B.
For example, if A = {a, b, c, d) and B = {c, d, e, f}, then
A
B = {c, d}.
Page 3
SETS
2.2 Set-Builder Form
In this form, we write a variable (say x) representing any
member of the set followed by a property satisfied by each
member of the set.
For example, the set A of all prime numbers less than 10 in
the set-builder form is written as
A = {x | x is a prime number less that 10}
The symbol '|' stands for the words 'such that'. Sometimes,
we use the symbol ':' in place of the symbol '|'.
3. TYPES OF SETS
3.1 Empty Set or Null Set
A set which has no element is called the null set or empty
set. It is denoted by the symbol .
For example, each of the following is a null set :
(a) The set of all real numbers whose square is –1.
(b) The set of all rational numbers whose square is 2.
(c) The set of all those integers that are both even and odd.
A set consisting of atleast one element is called a
non-empty set.
3.2 Singleton Set
A set having only one element is called singleton set.
For example, {0} is a singleton set, whose only member is 0.
3.3 Finite and Infinite Set
A set which has finite number of elements is called a finite
set. Otherwise, it is called an infinite set.
For example, the set of all days in a week is a finite set
whereas the set of all integers, denoted by
{............ -2, -1, 0, 1, 2,...} or {x | x is an integer}, is an infinite set.
An empty set which has no element in a finite set A is
called empty of void or null set.
1. SET
A set is a collection of well-defined and well distinguished
objects of our perception or thought.
1.1 Notations
The sets are usually denoted by capital letters A, B, C, etc.
and the members or elements of the set are denoted by lower-
case letters a, b, c, etc. If x is a member of the set A, we write
x
A (read as 'x belongs to A') and if x is not a member of the
set A, we write x A (read as 'x does not belong to A,). If x
and y both belong to A, we write x, y
A.
2. REPRESENTATION OF A SET
Usually, sets are represented in the following two ways :
(i) Roster form or Tabular form
(ii) Set Builder form or Rule Method
2.1 Roster Form
In this form, we list all the member of the set within braces
(curly brackets) and separate these by commas. For example,
the set A of all odd natural numbers less that 10 in the Roster
form is written as :
A = {1, 3, 5, 7, 9}
(i) In roster form, every element of the set is listed
only once.
(ii) The order in which the elements are listed is
immaterial.
For example, each of the following sets denotes
the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}
SETS, RELATIONS & FUNCTIONS
SETS, RELATIONS & FUNCTIONS SETS, RELATIONS & FUNCTIONS
3.4 Cardinal Number
The number of elements in finite set is represented by n(A),
known as Cardinal number.
3.5 Equal Sets
Two sets A and B are said to be equals, written as A = B, if
every element of A is in B and every element of B is in A.
3.6 Equivalent Sets
Two finite sets A and B are said to be equivalent, if n
(A) = n (B). Clearly, equal sets are equivalent but equivalent
sets need not be equal.
For example, the sets A = { 4, 5, 3, 2} and B = {1, 6, 8, 9} are
equivalent but are not equal.
3.7 Subset
Let A and B be two sets. If every elements of A is an element
of B, then A is called a subset of B and we write A B or
B A (read as 'A is contained in B' or B contains A'). B is
called superset of A.
(i) Every set is a subset and a superset itself.
(ii) If A is not a subset of B, we write A B.
(iii) The empty set is the subset of every set.
(iv) If A is a set with n(A) = m, then the number of
subsets of A are 2
m
and the number of proper
subsets of A are 2
m
-1.
For example, let A = {3, 4}, then the subsets of A
are , {3}, {4}. {3, 4}. Here, n(A) = 2 and number
of subsets of A = 2
2
= 4. Also, {3}
{3,4}and {2,3}
{3, 4}
3.8 Power Set
The set of all subsets of a given set A is called the power set
of A and is denoted by P(A).
For example, if A = {1, 2, 3}, then
P(A) = { , {1}, {2}, {3}, {1,2} {1, 3}, {2, 3}, {1, 2, 3}}
Clearly, if A has n elements, then its power set P (A) contains
exactly 2
n
elements.
4. OPERATIONS ON SETS
4.1 Union of Two Sets
The union of two sets A and B, written as A B (read as 'A
union B'), is the set consisting of all the elements which are
either in A or in B or in both Thus,
A B = {x : x
A or x
B}
Clearly, x
A B x
A A or x
B, and
x A B x A A and x B.
For example, if A = {a, b, c, d} and B = {c, d, e, f}, then
A B = {a, b, c, d, e, f}
4.2 Intersection of Two sets
The intersection of two sets A and B, written as A
B
(read as ‘A ’ intersection ‘B’) is the set consisting of all the
common elements of A and B. Thus,
A
B = {x : x
A and x
B}
Clearly, x
A
B x
A A and x
B, and
x
A
B x
A A or x
B.
For example, if A = {a, b, c, d) and B = {c, d, e, f}, then
A
B = {c, d}.
SETS, RELATIONS & FUNCTIONS
4.3 Disjoint Sets
Two sets A and B are said to be disjoint, if A
B = , i.e. A A
and B have no element in common.
For example, if A = {1, 3, 5} and B = {2, 4, 6},
then A
B = , so A and B are disjoint sets.
4.4 Difference of Two Sets
If A and B are two sets, then their difference A - B is defined
as :
A - B = {x : x
A and x B}.
Similarly, B - A = {x : x
B and x A }.
For example, if A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 9} then A
- B = {2, 4} and B - A = {7, 9}.
Important Results
(a) A - B
B – A A
(b) The sets A - B , B - A and A B are disjoint sets
(c) A - B A and B - A B
(d) A - = A and A - A =
4.5 Symmetric Difference of Two Sets
The symmetric difference of two sets A and B , denoted by
A
B, is defined as
A
B = (A - B) (B - A).
For example, if A = {1,2,3,4,5} and B = {1, 3,5,7,9} then
A
B = (A - B) (B - A) = {2,4} {7,9} = {2,4,7,9}.
4.6 Complement of a Set
If U is a universal set and A is a subset of U, then the
complement of A is the set which contains those elements
of U, which are not contained in A and is denoted by
A'or A
c
. Thus,
A
c
= {x : x
U and x A} A}
For example, if U = {1,2,3,4 ...} and A {2,4,6,8,...}, then, A
c
=
{1,3,5,7, ...}
Important Results
a) U
c
= b)
c
= U c) A A A
c
= U
d) A
A
c
=
5. ALGEBRA OF SETS
1. For any set A , we have
a) A A = A A b) A
A = A A
2. For any set A, we have
c) A = A A d) A
=
e) A U = U f) A
U = A A
3. For any two sets A and B, we have
g) A B = B A A h) A
B = B
A A
4. For any three sets A, B and C, we have
i) A (B C) = (A B) C
j) A (B
C) = (A
B)
C
5. For any three sets A, B and C, we have
k) A (B
C) = (A B)
(A C)
l) A
(B C) = (A
B) (A
C)
6. If A is any set, we have (A
c
)
c
= A.
7. Demorgan's Laws For any three sets A, B and C, we have
m) (A B)
c
= A A
c
B
c
n) (A
B)
c
= A A
c
B
c
o) A - (B C) = (A - B)
( A - C)
p) A - (B
C) = (A - B) (A - C)
Page 4
SETS
2.2 Set-Builder Form
In this form, we write a variable (say x) representing any
member of the set followed by a property satisfied by each
member of the set.
For example, the set A of all prime numbers less than 10 in
the set-builder form is written as
A = {x | x is a prime number less that 10}
The symbol '|' stands for the words 'such that'. Sometimes,
we use the symbol ':' in place of the symbol '|'.
3. TYPES OF SETS
3.1 Empty Set or Null Set
A set which has no element is called the null set or empty
set. It is denoted by the symbol .
For example, each of the following is a null set :
(a) The set of all real numbers whose square is –1.
(b) The set of all rational numbers whose square is 2.
(c) The set of all those integers that are both even and odd.
A set consisting of atleast one element is called a
non-empty set.
3.2 Singleton Set
A set having only one element is called singleton set.
For example, {0} is a singleton set, whose only member is 0.
3.3 Finite and Infinite Set
A set which has finite number of elements is called a finite
set. Otherwise, it is called an infinite set.
For example, the set of all days in a week is a finite set
whereas the set of all integers, denoted by
{............ -2, -1, 0, 1, 2,...} or {x | x is an integer}, is an infinite set.
An empty set which has no element in a finite set A is
called empty of void or null set.
1. SET
A set is a collection of well-defined and well distinguished
objects of our perception or thought.
1.1 Notations
The sets are usually denoted by capital letters A, B, C, etc.
and the members or elements of the set are denoted by lower-
case letters a, b, c, etc. If x is a member of the set A, we write
x
A (read as 'x belongs to A') and if x is not a member of the
set A, we write x A (read as 'x does not belong to A,). If x
and y both belong to A, we write x, y
A.
2. REPRESENTATION OF A SET
Usually, sets are represented in the following two ways :
(i) Roster form or Tabular form
(ii) Set Builder form or Rule Method
2.1 Roster Form
In this form, we list all the member of the set within braces
(curly brackets) and separate these by commas. For example,
the set A of all odd natural numbers less that 10 in the Roster
form is written as :
A = {1, 3, 5, 7, 9}
(i) In roster form, every element of the set is listed
only once.
(ii) The order in which the elements are listed is
immaterial.
For example, each of the following sets denotes
the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}
SETS, RELATIONS & FUNCTIONS
SETS, RELATIONS & FUNCTIONS SETS, RELATIONS & FUNCTIONS
3.4 Cardinal Number
The number of elements in finite set is represented by n(A),
known as Cardinal number.
3.5 Equal Sets
Two sets A and B are said to be equals, written as A = B, if
every element of A is in B and every element of B is in A.
3.6 Equivalent Sets
Two finite sets A and B are said to be equivalent, if n
(A) = n (B). Clearly, equal sets are equivalent but equivalent
sets need not be equal.
For example, the sets A = { 4, 5, 3, 2} and B = {1, 6, 8, 9} are
equivalent but are not equal.
3.7 Subset
Let A and B be two sets. If every elements of A is an element
of B, then A is called a subset of B and we write A B or
B A (read as 'A is contained in B' or B contains A'). B is
called superset of A.
(i) Every set is a subset and a superset itself.
(ii) If A is not a subset of B, we write A B.
(iii) The empty set is the subset of every set.
(iv) If A is a set with n(A) = m, then the number of
subsets of A are 2
m
and the number of proper
subsets of A are 2
m
-1.
For example, let A = {3, 4}, then the subsets of A
are , {3}, {4}. {3, 4}. Here, n(A) = 2 and number
of subsets of A = 2
2
= 4. Also, {3}
{3,4}and {2,3}
{3, 4}
3.8 Power Set
The set of all subsets of a given set A is called the power set
of A and is denoted by P(A).
For example, if A = {1, 2, 3}, then
P(A) = { , {1}, {2}, {3}, {1,2} {1, 3}, {2, 3}, {1, 2, 3}}
Clearly, if A has n elements, then its power set P (A) contains
exactly 2
n
elements.
4. OPERATIONS ON SETS
4.1 Union of Two Sets
The union of two sets A and B, written as A B (read as 'A
union B'), is the set consisting of all the elements which are
either in A or in B or in both Thus,
A B = {x : x
A or x
B}
Clearly, x
A B x
A A or x
B, and
x A B x A A and x B.
For example, if A = {a, b, c, d} and B = {c, d, e, f}, then
A B = {a, b, c, d, e, f}
4.2 Intersection of Two sets
The intersection of two sets A and B, written as A
B
(read as ‘A ’ intersection ‘B’) is the set consisting of all the
common elements of A and B. Thus,
A
B = {x : x
A and x
B}
Clearly, x
A
B x
A A and x
B, and
x
A
B x
A A or x
B.
For example, if A = {a, b, c, d) and B = {c, d, e, f}, then
A
B = {c, d}.
SETS, RELATIONS & FUNCTIONS
4.3 Disjoint Sets
Two sets A and B are said to be disjoint, if A
B = , i.e. A A
and B have no element in common.
For example, if A = {1, 3, 5} and B = {2, 4, 6},
then A
B = , so A and B are disjoint sets.
4.4 Difference of Two Sets
If A and B are two sets, then their difference A - B is defined
as :
A - B = {x : x
A and x B}.
Similarly, B - A = {x : x
B and x A }.
For example, if A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 9} then A
- B = {2, 4} and B - A = {7, 9}.
Important Results
(a) A - B
B – A A
(b) The sets A - B , B - A and A B are disjoint sets
(c) A - B A and B - A B
(d) A - = A and A - A =
4.5 Symmetric Difference of Two Sets
The symmetric difference of two sets A and B , denoted by
A
B, is defined as
A
B = (A - B) (B - A).
For example, if A = {1,2,3,4,5} and B = {1, 3,5,7,9} then
A
B = (A - B) (B - A) = {2,4} {7,9} = {2,4,7,9}.
4.6 Complement of a Set
If U is a universal set and A is a subset of U, then the
complement of A is the set which contains those elements
of U, which are not contained in A and is denoted by
A'or A
c
. Thus,
A
c
= {x : x
U and x A} A}
For example, if U = {1,2,3,4 ...} and A {2,4,6,8,...}, then, A
c
=
{1,3,5,7, ...}
Important Results
a) U
c
= b)
c
= U c) A A A
c
= U
d) A
A
c
=
5. ALGEBRA OF SETS
1. For any set A , we have
a) A A = A A b) A
A = A A
2. For any set A, we have
c) A = A A d) A
=
e) A U = U f) A
U = A A
3. For any two sets A and B, we have
g) A B = B A A h) A
B = B
A A
4. For any three sets A, B and C, we have
i) A (B C) = (A B) C
j) A (B
C) = (A
B)
C
5. For any three sets A, B and C, we have
k) A (B
C) = (A B)
(A C)
l) A
(B C) = (A
B) (A
C)
6. If A is any set, we have (A
c
)
c
= A.
7. Demorgan's Laws For any three sets A, B and C, we have
m) (A B)
c
= A A
c
B
c
n) (A
B)
c
= A A
c
B
c
o) A - (B C) = (A - B)
( A - C)
p) A - (B
C) = (A - B) (A - C)
SETS, RELATIONS & FUNCTIONS
Important Results on Operations on Sets
(i) A A B, B A B, A A
B A, A A
B B
(ii) A - B = A
B
c
(iii) (A - B) B = A B
(iv) (A - B)
B = (v) A B
B
c
A
c
(vi) A - B = B
c
- A
c
(vii) (A B)
(A B
c
) = A A
(viii) A B = (A - B) (B - A) (A
B)
(ix) A - (A - B) = A
B
(x) A - B = B - A
A = B (xi) AB = A A
B
A = B
(xii) A
(B
C) = (A
B)
(A
C)
Example – 1
Write the set of all positive integers whose cube is odd.
Sol. The elements of the required set are not even.
[
Cube of an even integer is also an even integer]
Moreover, the cube of a positive odd integer is a positive
odd integer.
The elements of the required set are all positive odd integers.
Hence, the required set, in the set builder form, is :
2k 1 : k 0, k Z .
Example – 2
Write the set
1 2 3 4 5 6 7
, , , , , ,
2 3 4 5 6 7 8
in the set
builder form.
Sol. In each element of the given set the denominator is one
more than the numerator.
Also the numerators are from 1 to 7.
Hence the set builder form of the given set is :
x : x n / n 1, n N and 1 n 7 .
Example – 3
Write the set {x : x is a positive integer and x
2
< 30} in the
roster form.
Sol. The squares of positive integers whose squares are less
than 30 are : 1, 2, 3, 4, 5.
Hence the given set, in roster form, is {1, 2, 3, 4, 5}.
Example – 4
Write the set {0, 1, 4, 9, 16, .......} in set builder form.
Sol. The elements of the given set are squares of integers :
0, 1, 2, 3, 4, .......
Hence the given set, in set builder form, is {x
2
: xZ}.
Example – 5
State which of the following sets are finite and which are
infinite
(i) A = {x : x N and x
2
– 3x + 2 = 0}
(ii) B = {x : x N and x
2
= 9}
(iii) C = {x : x N and x is even}
(iv) D = {x : x N and 2x – 3 = 0}.
Sol. (i) A = {1, 2}.
[
x
2
– 3x + 2 = 0 (x – 1) (x – 2) = 0 x = 1, 2]
Hence A is finite.
(ii) B = {3}.
[
x
2
= 9 x = + 3. But 3 N]
Hence B is finite.
(iii) C = {2, 4, 6, ......}
Hence C is infinite.
(iv) D = .
3
2x 3 0 x N
2
Hence D is finite.
Page 5
SETS
2.2 Set-Builder Form
In this form, we write a variable (say x) representing any
member of the set followed by a property satisfied by each
member of the set.
For example, the set A of all prime numbers less than 10 in
the set-builder form is written as
A = {x | x is a prime number less that 10}
The symbol '|' stands for the words 'such that'. Sometimes,
we use the symbol ':' in place of the symbol '|'.
3. TYPES OF SETS
3.1 Empty Set or Null Set
A set which has no element is called the null set or empty
set. It is denoted by the symbol .
For example, each of the following is a null set :
(a) The set of all real numbers whose square is –1.
(b) The set of all rational numbers whose square is 2.
(c) The set of all those integers that are both even and odd.
A set consisting of atleast one element is called a
non-empty set.
3.2 Singleton Set
A set having only one element is called singleton set.
For example, {0} is a singleton set, whose only member is 0.
3.3 Finite and Infinite Set
A set which has finite number of elements is called a finite
set. Otherwise, it is called an infinite set.
For example, the set of all days in a week is a finite set
whereas the set of all integers, denoted by
{............ -2, -1, 0, 1, 2,...} or {x | x is an integer}, is an infinite set.
An empty set which has no element in a finite set A is
called empty of void or null set.
1. SET
A set is a collection of well-defined and well distinguished
objects of our perception or thought.
1.1 Notations
The sets are usually denoted by capital letters A, B, C, etc.
and the members or elements of the set are denoted by lower-
case letters a, b, c, etc. If x is a member of the set A, we write
x
A (read as 'x belongs to A') and if x is not a member of the
set A, we write x A (read as 'x does not belong to A,). If x
and y both belong to A, we write x, y
A.
2. REPRESENTATION OF A SET
Usually, sets are represented in the following two ways :
(i) Roster form or Tabular form
(ii) Set Builder form or Rule Method
2.1 Roster Form
In this form, we list all the member of the set within braces
(curly brackets) and separate these by commas. For example,
the set A of all odd natural numbers less that 10 in the Roster
form is written as :
A = {1, 3, 5, 7, 9}
(i) In roster form, every element of the set is listed
only once.
(ii) The order in which the elements are listed is
immaterial.
For example, each of the following sets denotes
the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}
SETS, RELATIONS & FUNCTIONS
SETS, RELATIONS & FUNCTIONS SETS, RELATIONS & FUNCTIONS
3.4 Cardinal Number
The number of elements in finite set is represented by n(A),
known as Cardinal number.
3.5 Equal Sets
Two sets A and B are said to be equals, written as A = B, if
every element of A is in B and every element of B is in A.
3.6 Equivalent Sets
Two finite sets A and B are said to be equivalent, if n
(A) = n (B). Clearly, equal sets are equivalent but equivalent
sets need not be equal.
For example, the sets A = { 4, 5, 3, 2} and B = {1, 6, 8, 9} are
equivalent but are not equal.
3.7 Subset
Let A and B be two sets. If every elements of A is an element
of B, then A is called a subset of B and we write A B or
B A (read as 'A is contained in B' or B contains A'). B is
called superset of A.
(i) Every set is a subset and a superset itself.
(ii) If A is not a subset of B, we write A B.
(iii) The empty set is the subset of every set.
(iv) If A is a set with n(A) = m, then the number of
subsets of A are 2
m
and the number of proper
subsets of A are 2
m
-1.
For example, let A = {3, 4}, then the subsets of A
are , {3}, {4}. {3, 4}. Here, n(A) = 2 and number
of subsets of A = 2
2
= 4. Also, {3}
{3,4}and {2,3}
{3, 4}
3.8 Power Set
The set of all subsets of a given set A is called the power set
of A and is denoted by P(A).
For example, if A = {1, 2, 3}, then
P(A) = { , {1}, {2}, {3}, {1,2} {1, 3}, {2, 3}, {1, 2, 3}}
Clearly, if A has n elements, then its power set P (A) contains
exactly 2
n
elements.
4. OPERATIONS ON SETS
4.1 Union of Two Sets
The union of two sets A and B, written as A B (read as 'A
union B'), is the set consisting of all the elements which are
either in A or in B or in both Thus,
A B = {x : x
A or x
B}
Clearly, x
A B x
A A or x
B, and
x A B x A A and x B.
For example, if A = {a, b, c, d} and B = {c, d, e, f}, then
A B = {a, b, c, d, e, f}
4.2 Intersection of Two sets
The intersection of two sets A and B, written as A
B
(read as ‘A ’ intersection ‘B’) is the set consisting of all the
common elements of A and B. Thus,
A
B = {x : x
A and x
B}
Clearly, x
A
B x
A A and x
B, and
x
A
B x
A A or x
B.
For example, if A = {a, b, c, d) and B = {c, d, e, f}, then
A
B = {c, d}.
SETS, RELATIONS & FUNCTIONS
4.3 Disjoint Sets
Two sets A and B are said to be disjoint, if A
B = , i.e. A A
and B have no element in common.
For example, if A = {1, 3, 5} and B = {2, 4, 6},
then A
B = , so A and B are disjoint sets.
4.4 Difference of Two Sets
If A and B are two sets, then their difference A - B is defined
as :
A - B = {x : x
A and x B}.
Similarly, B - A = {x : x
B and x A }.
For example, if A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 9} then A
- B = {2, 4} and B - A = {7, 9}.
Important Results
(a) A - B
B – A A
(b) The sets A - B , B - A and A B are disjoint sets
(c) A - B A and B - A B
(d) A - = A and A - A =
4.5 Symmetric Difference of Two Sets
The symmetric difference of two sets A and B , denoted by
A
B, is defined as
A
B = (A - B) (B - A).
For example, if A = {1,2,3,4,5} and B = {1, 3,5,7,9} then
A
B = (A - B) (B - A) = {2,4} {7,9} = {2,4,7,9}.
4.6 Complement of a Set
If U is a universal set and A is a subset of U, then the
complement of A is the set which contains those elements
of U, which are not contained in A and is denoted by
A'or A
c
. Thus,
A
c
= {x : x
U and x A} A}
For example, if U = {1,2,3,4 ...} and A {2,4,6,8,...}, then, A
c
=
{1,3,5,7, ...}
Important Results
a) U
c
= b)
c
= U c) A A A
c
= U
d) A
A
c
=
5. ALGEBRA OF SETS
1. For any set A , we have
a) A A = A A b) A
A = A A
2. For any set A, we have
c) A = A A d) A
=
e) A U = U f) A
U = A A
3. For any two sets A and B, we have
g) A B = B A A h) A
B = B
A A
4. For any three sets A, B and C, we have
i) A (B C) = (A B) C
j) A (B
C) = (A
B)
C
5. For any three sets A, B and C, we have
k) A (B
C) = (A B)
(A C)
l) A
(B C) = (A
B) (A
C)
6. If A is any set, we have (A
c
)
c
= A.
7. Demorgan's Laws For any three sets A, B and C, we have
m) (A B)
c
= A A
c
B
c
n) (A
B)
c
= A A
c
B
c
o) A - (B C) = (A - B)
( A - C)
p) A - (B
C) = (A - B) (A - C)
SETS, RELATIONS & FUNCTIONS
Important Results on Operations on Sets
(i) A A B, B A B, A A
B A, A A
B B
(ii) A - B = A
B
c
(iii) (A - B) B = A B
(iv) (A - B)
B = (v) A B
B
c
A
c
(vi) A - B = B
c
- A
c
(vii) (A B)
(A B
c
) = A A
(viii) A B = (A - B) (B - A) (A
B)
(ix) A - (A - B) = A
B
(x) A - B = B - A
A = B (xi) AB = A A
B
A = B
(xii) A
(B
C) = (A
B)
(A
C)
Example – 1
Write the set of all positive integers whose cube is odd.
Sol. The elements of the required set are not even.
[
Cube of an even integer is also an even integer]
Moreover, the cube of a positive odd integer is a positive
odd integer.
The elements of the required set are all positive odd integers.
Hence, the required set, in the set builder form, is :
2k 1 : k 0, k Z .
Example – 2
Write the set
1 2 3 4 5 6 7
, , , , , ,
2 3 4 5 6 7 8
in the set
builder form.
Sol. In each element of the given set the denominator is one
more than the numerator.
Also the numerators are from 1 to 7.
Hence the set builder form of the given set is :
x : x n / n 1, n N and 1 n 7 .
Example – 3
Write the set {x : x is a positive integer and x
2
< 30} in the
roster form.
Sol. The squares of positive integers whose squares are less
than 30 are : 1, 2, 3, 4, 5.
Hence the given set, in roster form, is {1, 2, 3, 4, 5}.
Example – 4
Write the set {0, 1, 4, 9, 16, .......} in set builder form.
Sol. The elements of the given set are squares of integers :
0, 1, 2, 3, 4, .......
Hence the given set, in set builder form, is {x
2
: xZ}.
Example – 5
State which of the following sets are finite and which are
infinite
(i) A = {x : x N and x
2
– 3x + 2 = 0}
(ii) B = {x : x N and x
2
= 9}
(iii) C = {x : x N and x is even}
(iv) D = {x : x N and 2x – 3 = 0}.
Sol. (i) A = {1, 2}.
[
x
2
– 3x + 2 = 0 (x – 1) (x – 2) = 0 x = 1, 2]
Hence A is finite.
(ii) B = {3}.
[
x
2
= 9 x = + 3. But 3 N]
Hence B is finite.
(iii) C = {2, 4, 6, ......}
Hence C is infinite.
(iv) D = .
3
2x 3 0 x N
2
Hence D is finite.
SETS, RELATIONS & FUNCTIONS
Example – 6
Which of the following are empty (null) sets ?
(i) Set of odd natural numbers divisible by 2
(ii) {x : 3 < x < 4, x N}
(iii) {x : x
2
= 25 and x is an odd integer}
(iv) [x : x
2
– 2 = 0 and x is rational]
(v) {x : x is common point of any two parallel lines}.
Sol. (i) Since there is no odd natural number, which is divisible
by 2.
it is an empty set.
(ii) Since there is no natural number between 3 and 4.
it is an empty set.
(iii) Now x
2
= 25 x = + 5, both are odd.
The set {– 5, 5} is non-emptry.
(iv) Since there is no rational number whose square is 2,
the given set is an empty set.
(v) Since any two parallel lines have no common point,
the given set is an empty set.
Example – 7
Find the pairs of equal sets from the following sets, if any,
giving reasons :
A = {0}, B = {x : x > 15 and x < 5},
C = {x : x – 5 = 0}, D = {x : x
2
= 25},
E = {x : x is a positive integral root of the equation
x
2
– 2x – 15 = 0}.
Sol. Here we have,
A = {0}
B =
[
There is no number, which is greater than 15 and less
than 5]
C = {5} [
x – 5 = 0 x = 5]
D = {– 5, 5} [
x
2
= 25 x = + 5]
and E = {5}.
[
x
2
– 2x – 15 = 0 (x – 5) (x + 3) = 0 x = 5, – 3. Out of
these two,
5 is positive integral]
Clearly C = E.
Example – 8
Are the following pairs of sets equal ? Give reasons.
(i) A = {1, 2}, B = {x : x is a solution of x
2
+ 3x + 2 = 0}
(ii) A = {x : x is a letter in the word FOLLOW},
B = {y : y is a letter in the word WOLF}.
Sol. (i) A = {1, 2}, B = {–2, –1}
[
x
2
+ 3x + 2 = 0 !!!(x + 2) (x + 1) = 0 !!!x = –2, —1]
Clearly
A B.
(ii) A = {F, O, L, L, O, W} = {F, O, L, W}
B = {W, O, L, F} = {F, O, L, W}.
Clearly A = B.
Example – 9
Let A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, C = {6, 7, 8, 9} and
D = {7, 8, 9, 10}. Find :
(a) (i)
A B
(ii)
B D
(iii)
A B C
(iv)
B C D
(b) (i)
A B
(ii)
B D
(iii)
A B C.
Sol. (a) (i)
A B
= {1, 2, 3, 4, 5}
{3, 4, 5, 6, 7}
= {1, 2, 3, 4, 5, 6, 7}.
(ii) B D = {3, 4, 5, 6, 7}
{7, 8, 9, 10}
= {3, 4, 5, 6, 7, 8, 9, 10}.
(iii)
A B C
= {1, 2, 3, 4, 5}
{3, 4, 5, 6, 7}
{6, 7, 8, 9}.
= {1, 2, 3, 4, 5, 6, 7}
{6, 7, 8, 9} = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
(iv)
B C D
= {3, 4, 5, 6, 7}
{6, 7, 8, 9}
{7, 8, 9, 10}.
= {3, 4, 5, 6, 7, 8, 9}
{7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}.
(b) (i) A B = {1, 2, 3, 4, 5}
{3, 4, 5, 6, 7} = {3, 4, 5}.
(ii) B D = {3, 4, 5, 6, 7}
{7, 8, 9, 10} = {7}.
(iii) A B C = {1, 2, 3, 4, 5}
{3, 4, 5, 6, 7}
{6, 7, 8, 9} = {3,
4, 5}
{6, 7, 8, 9} = .
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