Page 46 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Now what about the countability of the set R of real numbers?
Suppose that R is countable. Then an interval ]0, 1[, being an infinite subset of R, must be
countable. But then, we have already proved that the set ]0, 1[ is not countable. Hence R cannot
be countable.
Thus even by the method of countability of sets, we have established the much desired distinction
between Q and R in the sense that Q is countable but R is not countable.
Also, we observe that although R is not countable, yet it contains subsets which are countable.
For example R has subsets as Q, Z and N which are countable. At the same time R is an infinite
set. We sum up this observation in the form of the following theorem:
Theorem 8: Every infinite set contains a denumerable set.

Proof: Let S be an infinite set. Consider some element of S. Denote it by n . Consider the set
1
S – {a }. Now pick up an element from the new set and denote it by a .
1 2
Consider the set
S = {a , a }.
1 2
Proceeding in this way, having chosen a , you can have the set
k–l
S = {a , a , ..... a ,}.
1 2 k–1
This set is always non-empty because S is an infinite set. Hence, we can choose an element in this
set. Denote the element by a . This can be done for each k  N. Thus the set
k
{a , a , ....., a , .....}
1 2 k
is a denumerable subset of S and hence a countable subset of S. This proves the theorem.
The importance of this theorem is that it leads us to an interesting area of Cardinality of sets by
which we can determine and compare the relative sizes of various infinite sets,
This, however, is beyond the scope of this course.

Self Assessment

Fill in the blanks:
1. Let E denote the set of all even natural numbers. Then the mapping f: N  E defined as
.......................... is a one-one correspondence. Hence, the set E of even natural numbers is a
denumerable set and hence a countable set.
2. Let D denote the set of all odd integers and E the set of even integers. Then the, mapping
f: E  D , defined as .......................... is a one-one correspondence. Thus E ~ D, But, E – N,
therefore D – N. Hence D is a denumerable set and hence a countable set.
3. Every ...................... of a denumerable set is denumerable.

4. The set of all rational numbers between [0, 1] is ...................
5. The set of all .................... numbers is countable.

2.4 Summary

 We have discussed the order-relations (inequalities) in the set R of real numbers. Given
any two real numbers x and y, either x > y or x = y or x < y.

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