Revision Notes: Sets | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


 
1 
 
SETS, RELATIONS AND FUNCTIONS 
 
Set: 
A set is a collection of well-defined objects i.e. the objects follow 
a given rule or rules.  
 
Elements of a set: 
The members of a set are called its elements. If an element x is in 
set A, we say that x belongs to A and write x ? A. If the element 
x is not in A then we write x ? A. 
Examples of sets: 
1. The set of vowels in the alphabet of English language. 
 2. The set of all points on a particular line. 
  
Some special sets: 
(i)Finite and infinite sets: 
A set A is finite if it contains only a finite number of elements; 
we can find the exact number of elements in the set. Otherwise, 
the set is said to be an infinite set. 
Example: 
 Q = set of all rational numbers = 
p
:p, q Z,q 0
q
??
??
??
??
 
 R = set of all real numbers = {x: x is a rational and an 
irrational number} 
 C = set of all complex numbers = ? ? x iy; x,y R ?? 
 
 
 
Page 2


 
1 
 
SETS, RELATIONS AND FUNCTIONS 
 
Set: 
A set is a collection of well-defined objects i.e. the objects follow 
a given rule or rules.  
 
Elements of a set: 
The members of a set are called its elements. If an element x is in 
set A, we say that x belongs to A and write x ? A. If the element 
x is not in A then we write x ? A. 
Examples of sets: 
1. The set of vowels in the alphabet of English language. 
 2. The set of all points on a particular line. 
  
Some special sets: 
(i)Finite and infinite sets: 
A set A is finite if it contains only a finite number of elements; 
we can find the exact number of elements in the set. Otherwise, 
the set is said to be an infinite set. 
Example: 
 Q = set of all rational numbers = 
p
:p, q Z,q 0
q
??
??
??
??
 
 R = set of all real numbers = {x: x is a rational and an 
irrational number} 
 C = set of all complex numbers = ? ? x iy; x,y R ?? 
 
 
 
 
2 
 
(ii) Null set:  
A set which does not contain any element is called a null set and 
is denoted by ?. A null set is also called an empty set. 
 
(iii) Singleton set: 
A set which contains only one element is called a singleton set. 
 
 (viii) Power set: 
The power set of a set A is the set of all of its subsets, and is 
denoted by  ? ? PA e.g. if ? ? A 4, 5, 6 ? then
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
P A , 4 , 5 , 6 , 4, 5 , 5, 6 , 4, 5, 6 ?? . 
Note: The null set ? and set A are always elements of ? ? PA . 
 
Theorem: If a finite set has n elements, then the power set of A 
has 
n
2 elements. 
Operations on sets: 
The operations on sets, by which sets can be combined to produce 
new sets. 
(i) Union of sets: 
The union of two set A and B is defined as the set of all 
elements which are either in A or in B or in both. The union of 
two sets is written asAB ? ;  
 
(ii)  Intersection of sets: 
(i) The intersection of two sets A and B is defined as the set of 
those elements which are in both A and B and is written as 
? ? A B x : x A and x B ? ? ? ? 
 
Page 3


 
1 
 
SETS, RELATIONS AND FUNCTIONS 
 
Set: 
A set is a collection of well-defined objects i.e. the objects follow 
a given rule or rules.  
 
Elements of a set: 
The members of a set are called its elements. If an element x is in 
set A, we say that x belongs to A and write x ? A. If the element 
x is not in A then we write x ? A. 
Examples of sets: 
1. The set of vowels in the alphabet of English language. 
 2. The set of all points on a particular line. 
  
Some special sets: 
(i)Finite and infinite sets: 
A set A is finite if it contains only a finite number of elements; 
we can find the exact number of elements in the set. Otherwise, 
the set is said to be an infinite set. 
Example: 
 Q = set of all rational numbers = 
p
:p, q Z,q 0
q
??
??
??
??
 
 R = set of all real numbers = {x: x is a rational and an 
irrational number} 
 C = set of all complex numbers = ? ? x iy; x,y R ?? 
 
 
 
 
2 
 
(ii) Null set:  
A set which does not contain any element is called a null set and 
is denoted by ?. A null set is also called an empty set. 
 
(iii) Singleton set: 
A set which contains only one element is called a singleton set. 
 
 (viii) Power set: 
The power set of a set A is the set of all of its subsets, and is 
denoted by  ? ? PA e.g. if ? ? A 4, 5, 6 ? then
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
P A , 4 , 5 , 6 , 4, 5 , 5, 6 , 4, 5, 6 ?? . 
Note: The null set ? and set A are always elements of ? ? PA . 
 
Theorem: If a finite set has n elements, then the power set of A 
has 
n
2 elements. 
Operations on sets: 
The operations on sets, by which sets can be combined to produce 
new sets. 
(i) Union of sets: 
The union of two set A and B is defined as the set of all 
elements which are either in A or in B or in both. The union of 
two sets is written asAB ? ;  
 
(ii)  Intersection of sets: 
(i) The intersection of two sets A and B is defined as the set of 
those elements which are in both A and B and is written as 
? ? A B x : x A and x B ? ? ? ? 
 
 
3 
 
(ii) The intersection of n sets 
1 2 n
A , A ........A is written as  
 ? ?
n
i 1 2 3 n i
i1
A A A A ......... A x : x A for all i, 1 i n
?
? ? ? ? ? ? ? ? ? . 
 
Disjoint sets: 
Two set A and B are said to be disjoint, if there is no element 
which is in both A and B, i.e. AB ? ? ?; 
? The properties of the complement of sets are known as 
DeMorgan laws, which are 
 (i)   
cc
A B B A ? ? ? 
 (ii)  ? ?
c
cc
A B A B ? ? ? 
 (iii) ? ?
c
cc
A B A B ? ? ? 
  
? If A and B are not disjoint, then  
 (i)   ? ? ? ? ? ? ? ? n A B n A n B n A B ? ? ? ? ? 
 (ii)  ? ? ? ? ? ? ? ? n A B n A B n B A n A B ? ? ? ? ? ? ? 
 (iii)  ? ? ? ? ? ? n A n A B n A B ? ? ? ? 
 (iv)  ? ? ? ? ? ? n B n B A n A B ? ? ? ? 
 
 
 (vi) Cartesian product of sets: 
Let a be an arbitrary element of a given set A i.e. aA ? and b be an 
arbitrary element of B i.e. bB ? . Then the pair ? ? a, b is an ordered 
pair. Obviously ? ? a, b ? ? b, a ? . The cartesian product of two sets A and 
B is defined as the set of ordered pairs ? ? a, b . The cartesian product 
is denoted by AB ? 
? ? ? ?
A B a, b ; a A, b B ? ? ? ? ? . 
Relation: 
Page 4


 
1 
 
SETS, RELATIONS AND FUNCTIONS 
 
Set: 
A set is a collection of well-defined objects i.e. the objects follow 
a given rule or rules.  
 
Elements of a set: 
The members of a set are called its elements. If an element x is in 
set A, we say that x belongs to A and write x ? A. If the element 
x is not in A then we write x ? A. 
Examples of sets: 
1. The set of vowels in the alphabet of English language. 
 2. The set of all points on a particular line. 
  
Some special sets: 
(i)Finite and infinite sets: 
A set A is finite if it contains only a finite number of elements; 
we can find the exact number of elements in the set. Otherwise, 
the set is said to be an infinite set. 
Example: 
 Q = set of all rational numbers = 
p
:p, q Z,q 0
q
??
??
??
??
 
 R = set of all real numbers = {x: x is a rational and an 
irrational number} 
 C = set of all complex numbers = ? ? x iy; x,y R ?? 
 
 
 
 
2 
 
(ii) Null set:  
A set which does not contain any element is called a null set and 
is denoted by ?. A null set is also called an empty set. 
 
(iii) Singleton set: 
A set which contains only one element is called a singleton set. 
 
 (viii) Power set: 
The power set of a set A is the set of all of its subsets, and is 
denoted by  ? ? PA e.g. if ? ? A 4, 5, 6 ? then
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
P A , 4 , 5 , 6 , 4, 5 , 5, 6 , 4, 5, 6 ?? . 
Note: The null set ? and set A are always elements of ? ? PA . 
 
Theorem: If a finite set has n elements, then the power set of A 
has 
n
2 elements. 
Operations on sets: 
The operations on sets, by which sets can be combined to produce 
new sets. 
(i) Union of sets: 
The union of two set A and B is defined as the set of all 
elements which are either in A or in B or in both. The union of 
two sets is written asAB ? ;  
 
(ii)  Intersection of sets: 
(i) The intersection of two sets A and B is defined as the set of 
those elements which are in both A and B and is written as 
? ? A B x : x A and x B ? ? ? ? 
 
 
3 
 
(ii) The intersection of n sets 
1 2 n
A , A ........A is written as  
 ? ?
n
i 1 2 3 n i
i1
A A A A ......... A x : x A for all i, 1 i n
?
? ? ? ? ? ? ? ? ? . 
 
Disjoint sets: 
Two set A and B are said to be disjoint, if there is no element 
which is in both A and B, i.e. AB ? ? ?; 
? The properties of the complement of sets are known as 
DeMorgan laws, which are 
 (i)   
cc
A B B A ? ? ? 
 (ii)  ? ?
c
cc
A B A B ? ? ? 
 (iii) ? ?
c
cc
A B A B ? ? ? 
  
? If A and B are not disjoint, then  
 (i)   ? ? ? ? ? ? ? ? n A B n A n B n A B ? ? ? ? ? 
 (ii)  ? ? ? ? ? ? ? ? n A B n A B n B A n A B ? ? ? ? ? ? ? 
 (iii)  ? ? ? ? ? ? n A n A B n A B ? ? ? ? 
 (iv)  ? ? ? ? ? ? n B n B A n A B ? ? ? ? 
 
 
 (vi) Cartesian product of sets: 
Let a be an arbitrary element of a given set A i.e. aA ? and b be an 
arbitrary element of B i.e. bB ? . Then the pair ? ? a, b is an ordered 
pair. Obviously ? ? a, b ? ? b, a ? . The cartesian product of two sets A and 
B is defined as the set of ordered pairs ? ? a, b . The cartesian product 
is denoted by AB ? 
? ? ? ?
A B a, b ; a A, b B ? ? ? ? ? . 
Relation: 
 
4 
 
Let A and B be two sets. A relation R from the set A to set B is a 
subset of the Cartesian productAB ? . Further, if ? ? x, y R ? , then we say 
that x is R-related to y and write this relation as x R y. Hence
? ? ? ?
R x, y ;x A, y B, x R y ?? . 
 
Domain and Range of a relation: Let R be a relation defined 
from a A set to a set B, i.e.R A B ?? . Then the set of all first elements 
of the ordered pairs in R is called the domain of R. The set of all 
second elements of the ordered pairs in R is called the range of R. 
That is,  
D = domain of ? ? ? ?
R x : x, y R ?? or
? ? ? ?
x : x A and x, y R ?? , 
R
?
 = range of ? ? ? ?
R y : x, y R ?? or
? ? ? ?
y : y Band x, y R ?? . 
Clearly DA ? and
*
RB ? . 
 
FUNCTIONS: 
A mapping f: X ? Y is said to be a function if each element in 
the set X has its image in set Y. Every element in set X 
should have one and only one image.  
Let f: R ? R where y = x
3
. Here for each x ? R we would have 
a unique value of y in the set R 
Set ‘X’ is called domain of the function ‘f’. 
Set ‘Y’ is called the co-domain of the function ‘f’. 
 
Page 5


 
1 
 
SETS, RELATIONS AND FUNCTIONS 
 
Set: 
A set is a collection of well-defined objects i.e. the objects follow 
a given rule or rules.  
 
Elements of a set: 
The members of a set are called its elements. If an element x is in 
set A, we say that x belongs to A and write x ? A. If the element 
x is not in A then we write x ? A. 
Examples of sets: 
1. The set of vowels in the alphabet of English language. 
 2. The set of all points on a particular line. 
  
Some special sets: 
(i)Finite and infinite sets: 
A set A is finite if it contains only a finite number of elements; 
we can find the exact number of elements in the set. Otherwise, 
the set is said to be an infinite set. 
Example: 
 Q = set of all rational numbers = 
p
:p, q Z,q 0
q
??
??
??
??
 
 R = set of all real numbers = {x: x is a rational and an 
irrational number} 
 C = set of all complex numbers = ? ? x iy; x,y R ?? 
 
 
 
 
2 
 
(ii) Null set:  
A set which does not contain any element is called a null set and 
is denoted by ?. A null set is also called an empty set. 
 
(iii) Singleton set: 
A set which contains only one element is called a singleton set. 
 
 (viii) Power set: 
The power set of a set A is the set of all of its subsets, and is 
denoted by  ? ? PA e.g. if ? ? A 4, 5, 6 ? then
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
P A , 4 , 5 , 6 , 4, 5 , 5, 6 , 4, 5, 6 ?? . 
Note: The null set ? and set A are always elements of ? ? PA . 
 
Theorem: If a finite set has n elements, then the power set of A 
has 
n
2 elements. 
Operations on sets: 
The operations on sets, by which sets can be combined to produce 
new sets. 
(i) Union of sets: 
The union of two set A and B is defined as the set of all 
elements which are either in A or in B or in both. The union of 
two sets is written asAB ? ;  
 
(ii)  Intersection of sets: 
(i) The intersection of two sets A and B is defined as the set of 
those elements which are in both A and B and is written as 
? ? A B x : x A and x B ? ? ? ? 
 
 
3 
 
(ii) The intersection of n sets 
1 2 n
A , A ........A is written as  
 ? ?
n
i 1 2 3 n i
i1
A A A A ......... A x : x A for all i, 1 i n
?
? ? ? ? ? ? ? ? ? . 
 
Disjoint sets: 
Two set A and B are said to be disjoint, if there is no element 
which is in both A and B, i.e. AB ? ? ?; 
? The properties of the complement of sets are known as 
DeMorgan laws, which are 
 (i)   
cc
A B B A ? ? ? 
 (ii)  ? ?
c
cc
A B A B ? ? ? 
 (iii) ? ?
c
cc
A B A B ? ? ? 
  
? If A and B are not disjoint, then  
 (i)   ? ? ? ? ? ? ? ? n A B n A n B n A B ? ? ? ? ? 
 (ii)  ? ? ? ? ? ? ? ? n A B n A B n B A n A B ? ? ? ? ? ? ? 
 (iii)  ? ? ? ? ? ? n A n A B n A B ? ? ? ? 
 (iv)  ? ? ? ? ? ? n B n B A n A B ? ? ? ? 
 
 
 (vi) Cartesian product of sets: 
Let a be an arbitrary element of a given set A i.e. aA ? and b be an 
arbitrary element of B i.e. bB ? . Then the pair ? ? a, b is an ordered 
pair. Obviously ? ? a, b ? ? b, a ? . The cartesian product of two sets A and 
B is defined as the set of ordered pairs ? ? a, b . The cartesian product 
is denoted by AB ? 
? ? ? ?
A B a, b ; a A, b B ? ? ? ? ? . 
Relation: 
 
4 
 
Let A and B be two sets. A relation R from the set A to set B is a 
subset of the Cartesian productAB ? . Further, if ? ? x, y R ? , then we say 
that x is R-related to y and write this relation as x R y. Hence
? ? ? ?
R x, y ;x A, y B, x R y ?? . 
 
Domain and Range of a relation: Let R be a relation defined 
from a A set to a set B, i.e.R A B ?? . Then the set of all first elements 
of the ordered pairs in R is called the domain of R. The set of all 
second elements of the ordered pairs in R is called the range of R. 
That is,  
D = domain of ? ? ? ?
R x : x, y R ?? or
? ? ? ?
x : x A and x, y R ?? , 
R
?
 = range of ? ? ? ?
R y : x, y R ?? or
? ? ? ?
y : y Band x, y R ?? . 
Clearly DA ? and
*
RB ? . 
 
FUNCTIONS: 
A mapping f: X ? Y is said to be a function if each element in 
the set X has its image in set Y. Every element in set X 
should have one and only one image.  
Let f: R ? R where y = x
3
. Here for each x ? R we would have 
a unique value of y in the set R 
Set ‘X’ is called domain of the function ‘f’. 
Set ‘Y’ is called the co-domain of the function ‘f’. 
 
 
5 
 
 
Algebra of Functions: 
Let us consider two functions,  
f: D
1
?R and g: D
2
?R .  We describe functions f + g, f - g, f.g 
and f/g as follows: 
? f + g : D ?R is a function defined by  
? (f + g)x = f(x)+g(x)       where D = D
1
?D
2
 
? f – g : D ? R is a function defined by  
? (f – g)x = f(x) –g(x) where D = D
1
?D
2
 
? f.g: D ?R is  a function defined by 
? by (f.g)x = f(x). g(x) where D = D
1
?D
2
 
? f/g: D ?R is  a function defined by 
? (f/g)x =
f(x)
g(x)
 where D = {x : x ? D
1
? D
2
, g(x) ?0} 
 
TYPE OF FUNCTION 
 
One-One and Many-One Functions:  
When every element of domain of a function has a distinct image 
in the co-domain, the function is said to be One-One. If there are 
at least two elements of the domain whose images are the same, 
the function is known as Many-One. 
 
Onto and Into Functions: 
For every point y in b, there is some point x in A such the  
f(x) = y. It is called onto function. When the codomain y which 
is not an image of any element in the domain x, then function is 
onto.