B.Sc Maths -V Semester Countable and Uncountable Sets Presented by Mrs.D.Renuga,Asst.Prof. in Mathematics,SAC Women's College,Cumbum

1. PRESENTED BY
MRS.D.RENUGA,M.SC.,M.PHIL.,M.ED.,
ASSISTANT PROFESSOR OF MATHEMATICS
SAC Women’s College,Cumbum.
E-mail: renugakannan238@gmail.com

2.  PRELIMINARIES
 FINITE SETS
 INFINITE SETS
 EQUIVALENT SETS
 COUNTABLE SETS
 COUNTABLY INFINITE SETS
 UNCOUNTABLE SETS
 DIFFERENCE BETWEEN CIS AND
UNCOUNTABLE SETS
 IMPORTANT RESULTS

3. NOTATIONS OF SET THEORY
 A B
 A B
 A B
 A
 A – B
 A X B
 f: A B
 The empty set which contains no
element is denoted by



c


4. DEFINITION
A set is said to finite if it contains finite
number of elements or ‘n’ number of
elements.
Example: A={1,2,3,4,……….100}
Cardinality of A :
n(A)=100

5. DEFINITION
A set which is not a finite that
set is called infinite set. Or Set having
infinite number of elements.
Example:1.Real Numbers,intervals etc.
2.Set B = {1,2,3,4,……….}
Cardinality of B :
n(B) =
 

6. DEFINITION
Two sets A and B are said to be
equivalent if there exists a bijection f from A
to B
Example: Let A = N and B ={2,4,6….,2n,….}
Then f:A B defined by f(n)=2n is a
bijection.
Hence A is equivalent to B even though A
has actually ‘more’ elements than B.



7. DEFINITION
A set is countable if either the set is
finite or we are able to put elements of the
set in order just like natural numbers are in
order.
For example: 1.N,Q are countable.
2.Prime numbers less than 20
A={2,3,5,7,11,13,17,19}
N 1,2,3,4, 5, 6, 7, 8
Here set A is countable finite.
cardinality of A ,ie.,n(A)= 8


8. DEFINITION
A set is countably infinite if its
elements can be put in one to one
Correspondence with the set of Natural
Numbers(set ~ 1N)
Example: f: N Z
N
x
x
x
f 2
;
2
)
( 





N
N
x
x 2
1
;
2
1 








Now, in this example we get a function which is bijective
from 1N Z Z is countable set
(Countably Infinite set)






9. DEFINITION
Set which is not finite and neither
equivalent to the set of Natural Number.
For example:
R,Qc ,any interval etc.

10. In Countably Infinite Set , one can count off
all elements in the set in such a way that,
even though the counting will take forever,
you will get to any particular element in a
finite amount of time.
For example:
Set = {0,1,-1,-2,-3,…..} is
Countably Infinite Set.
But uncountable set is so large, it cannot be
counted even if we kept counting forever.

11.  The set is a finite set,n( )=0
 The set primes less than 100 is finite
set P={2,3,5,…..97}
 The set of natural numbers is infinite
set.
 The set of all positive even numbers is
infinite set.
 Every finite set is countable.
 

12.  Empty set is countable.
 Every subset of a countable set is
countable.
 An uncountable set has both countable
and uncountable subsets
 If a set has uncountable subset then that
set is also uncountable.
 Every superset of an uncountable set is
uncountable.

13.  Every infinite set has countable subsets.
 Every infinite subset of a denumerable
set is denumerable.
 Every infinite subset of an uncountable
set is uncountable.
 Finite union of countable sets is
countable.
 Countable union of countable set is
countable.

14.  Intersection of Countable set is
countable.
 Finite product of countable sets is
countable.
 The Cartesian product of two countable
sets is countable.
 N X N is countable.
 Every interval is an uncountable set.

15.  The set of all rational numbers in [0,1] is
countable.
 The set of all real numbers in [0,1] is
uncountable.
 The set of all polynomials of degree less
than or equal to n , whose coefficients
are integers is countable.

16.  Let A is
countable
 If f:A B is one – one and A is countable,
then B can be countable or
uncountable
 If f:A B is one – one and B is countable,
then A is countable.
 If f:A B is one – one and A is uncountable,
 tthen B is countable.
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A 




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A
 B
A
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A
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B
A
 B
A

B
A

17. 

B
A
B
A
B
A

18.  The set of all rational numbers Q is
countable.
 The set of all prime numbers is
countable.
 The set of all irrational numbers Qc is
uncountable.
 The set of all real numbers R is
uncountable.
 The set of all complex numbers C is
uncountable.