JEE Mathematics/ Lakshmikanta Satapathy/ Fundamentals of set theory part 1/ Definition of set, Types of sets, empty set and infinite sets/ subset and power set/ Intervals as subsets of R

1. Physics Helpline
L K Satapathy
Set Theory 1

2. Physics Helpline
L K Satapathy
Definition : A set is a well defined collection of objects (also called elements).
2
{ : , 5}A x x n n N and n   
Set Theory 1
Representation of a set :
Roster form : In this form , all the elements of the set (without repetition) are listed.
Set builder form : In this form , a common defining property of all the elements of
the set is stated. The property should not depend on opinion.
Examples :
{1,4,9,16,25}A 
: , 1 5
1
n
B x x n N and n
n
 
     
 
2 3 4
, ,
3 4 5
B
 
  
 
Set builder form Roster form
For example , terms like ‘talented authors’ , ‘beautiful women’ depend on opinion
and hence can not be used as defining properties of sets .

3. Physics Helpline
L K Satapathy
Empty Set : A set which does not contain any element .
Set Theory 1
It is denoted by  or { }
2 5
:
3 6
A x x N and x
 
    
 
Example :
Since there is no natural number between and , A is an empty set.2
3
5
6
Finite Set : A set which has a finite number of elements
 2
: 3 2 0B x x N and x x    Example :
We have 2
3 2 0 ( 1)( 2) 0 1 2x x x x x or x         
 B = { 1 , 2 } which is a finite set having 2 elements
Infinite Set : A set which has infinite number of elements
Example :  :C x x N and x is odd 
It is an infinite set as there are infinite number of odd natural numbers

4. Physics Helpline
L K Satapathy
Set Theory 1
Question 1 : The set A = { x : x is a rational number and 1  x  2 } is
(a) Not a set (b) an empty set (c) a finite set (d) an infinite set
Answer The property { x : x is a rational number and 1  x  2 } is well defined.
Hence A is a set.  Option (a) is not correct
Again , between any two natural numbers , there are infinite rational numbers .
 The set A is an infinite set.  The correct option = (d) [Ans]
Question 2 : The set B = { x : x  N and 2x – 3 = 0 } is
(a) Not a set (b) an empty set (c) a finite set (d) an infinite set
Answer The property {x : x  N and 2x – 3 = 0} is well defined. Hence B is a set.
Now  No natural number satisfies the condition.
 The set B is an empty set.  The correct option = (b) [Ans]
3
2 3 0
2
x x x N     

5. Physics Helpline
L K Satapathy
Set Theory 1
Equal Sets : Sets A and B are said to be equal if every element of A is also an
element of B and every element of B is also an element of A .
In set notation , x  A  x  B and x  B  x  A
Example : A = { x : x is a prime number less than 6 }
B = { x : x is a prime factor of 60 }
 A = { 2 , 3 , 5 }
 A and B are equal sets
The prime numbers are 2 , 3 , 5 , 7 . . . .
 The prime numbers less than 6 are 2 , 3 , 5
Again , 60 = 2  2  3  5
 The prime factors of 60 are 2 , 3 and 5  B = { 2 , 3 , 5 }

6. Physics Helpline
L K Satapathy
Set Theory 1
Subsets :
Set A is said to be a subset of set B if every element of A is also an element of B.
In set notation , A  B if x  A  x  B
(i) Every set is a subset of itself .  A  A , B  B etc.
(ii) The empty set is a subset of every set    A ,   B etc.
(iii) If A  B and B  A
 Every element of A is also in B and every element of B is also in A
 A and B are equal sets
 A  B and B  A  A = B
(iv) If A  B but A  B , then A is a proper subset of B
 B is a superset of A . [  We write B  A ]

7. Physics Helpline
L K Satapathy
Set Theory 1
Subsets of the set (R) of Real Numbers :
(i) The Set of natural numbers N = { 1 , 2 , 3 , 4 , . . . . }
(ii) The Set of Integers Z = { . . . , – 4 , – 3 , – 2 , – 1 , 0 , 1 , 2 , 3 , 4 , . . . . }
(iii) The Set of Rational numbers : ; , 0
p
Q x x p q Z and q
q
 
    
 
(iv) The Set of Irrational numbers  :T x x R and x Q  
The Subsets Relations :
(i) N  Z , N  Q , N  R but N  T
(ii) Z  Q , Z  R but Z  T
(iii) Q  R , T  R
(iv) Q and T are disjoint sets ( No common element )

8. Physics Helpline
L K Satapathy
Set Theory 1
Intervals as Subsets of (R) :
( a , b ) = { x : a  x  b } is called an open interval ,
in which the end points a and b are not included.
[ a , b ] = { x : a  x  b } is called a closed interval ,
in which the end points a and b are included.
[ a , b ) = { x : a  x  b } is closed on the left and open
on right , in which a is included but b is not included.
( a , b ] = { x : a  x  b } is open on the left and closed
on right , in which b is included but a is not included.
a b
a b
a b
a b

9. Physics Helpline
L K Satapathy
Set Theory 1
Power Set of a Set :
The Set of all the subsets of a given Set is called the Power Set of that Set
Example: A = { a , b , c }  P(A) ={ ,{a},{b},{c},{a , b},{b, c},{a, c},{a ,b ,c}}
Rule : If A has n elements , then P(A) has elements.
The Power Set of a Set A is denoted as P(A)
2n
Proof : Number of subsets having no element ()
Number of subsets having 1 element
0
n
C
1
n
C
Number of subsets having 2 elements 2
n
C
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Number of subsets having n elements n
nC
 The number of elements of P(A) 0 1 2 . . . 2n n n n n
nC C C C     
We observe that P(A) has elements.3
2

10. Physics Helpline
L K Satapathy
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