The document discusses different types of sets in set theory, categorizing them as finite, infinite, countable, countably infinite, and uncountable. It provides definitions, examples, and important properties related to these sets, such as the equivalence of sets and cardinality. Additionally, it establishes key relationships among various types of sets, including unions, intersections, and products.

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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17. 
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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.