Real number system

1. Programme B.Sc.
Subject Mathematics
Semester 2
Unit 1, Real Number System
Session 2
Topic Countable and uncountable Some
standard theorems
Created by Dr.Syeda Rasheeda Parveen
ALGEBRA-II, and CALCULUS-II

2. Real Number System
In this unit, we shall study the following:
• Field of Real Numbers and
• Order Structure of real numbers
• Countable sets and uncountable sets
• Real line
• Bounded and unbounded sets-Supremum and
Infimum
• Order Completeness property
• Archimedean Property of real Numbers
• Intervals: Open sets, Closed sets and Countable sets.

3. Countable sets and uncountable
sets
I hope we are well acquainted with Set, Subset,
equality of sets, union and intersection of sets,
and union and intersection of an arbitrary family of
sets, Universal sets, functions, equivalent sets and
composition of structure, that is, addition and
multiplication
Now, we define countable and uncountable set

4. Countable sets and uncountable
sets
 Equivalent Sets : Two sets A & B are said to be
equivalent if there exists a one-one, onto mapping
from A to B. If A & B are equivalent, we denote this
relation by the symbol
Theorem: The relation “~” is an equivalence relation.
(I) ~ is reflexive.
(II) ~ is symmetric.
(III) ~ is transitive.


5. Countable sets and uncountable
sets
 (Denumerable or Enumerable or countable infinite)
Countable and Uncountable sets :
Definitions: (initial line segment of N). Let where N
is the set of natural numbers. The set
Is called an initial segment of N, determined by the
natural number n.
Example:
N8 = {1,2,3,4,5,6,7,8}
n N

{ / , }
n
N m m N m n
  

6. Countable sets and uncountable
sets
Definition (Finite set) : A set A is said to be a finite set,
if either or A~Nn for some .If a set then
n~Nn then is called the cardinal number of A.
Definition : (Infinite set): A set which is not finite is
called infinite set.
ie, a set A is called infinite set if and A is not
equivalent to Nn for any .
A   n N

A 
n N


7. Countable sets and uncountable sets
Denumerable Set :-
Definition : A set A set to be Denumerable, if ,A~N
where N is the set of natural numbers.
A denumerable set is also sometimes called
enumerable Or countably infinite set.

8. Countable sets and uncountable
sets
Countable Set :-
Definition :- A set A is said to be countable set if either
A is finite Or A denumerable.
The set A is said to be Uncountable set if A is not
countable.
I, e a set A is uncountable if A is not finite and A is not
equivalent to N.
Ex ample: The empty set is countable

9. Some Standard Theorems on
Countable and Uncountable:
Theorem 1: A subset of a finite set is finite.
Proof:
Theorem 2. A super set of am infinite set is also
infinite.
Proof:

10. Some Standard Theorems on
Countable and Uncountable:
Theorem 3 : Every subset of a countable set is
countable.
Proof:
Theorem 4 : Every super-set of an uncountable set is
uncountable.
Proof:
Theorem 5: Every infinity set has a denumerable
subset.

11. Some Standard Theorems on
Countable and Uncountable:
Theorem 6 : Every infinite set is equivalent to
one of its proper subsets.
Proof:
Theorem 7 : The Union of a countable
collection of countable sets is countable .
Proof:

12. Some Standard Theorems
Theorem 8 :- The set of all rational number is
countable.
Proof:
Theorem 9: Let A be a set and B is a countable set. If
there is Bijection between A & B, then A is also
countable.
Proof:
Theorem 10. :- If A and B be bijection countable sets,
then is also countable.
Proof:

13. Discussion
• show that the set Z of all integers is
countable.
• show that the set N×N is denumerable.

14. References
1) G K Ranganath College Mathematics
(S Chand& co).
2)G.K Ranganath College Mathematics Vol .1 (part 1)
(S Chand& co).
3) Dr.Syeda Rasheeda Parveen and Dr.Niyaz Bugum
Algebra II and Calculus II ( S.S.Bhavikatti Prakashana).