Page 39 - DMTH401_REAL ANALYSIS

Unit 2: Algebraic Structure and Countability

Thus y is an upper bound of S which does not belong to S. At the same time y is less than the Notes
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supremum of S. This is again absurd. Thus m > 2 is also not possible. Hence none of three
possibilities is true. This means there is something wrong with our supposition. In other words,
our supposition is false and therefore the set, S does not possess the suprernum in Q.
This justifies that the field Q of rational numbers is not order-complete.

Now you can also try a similar exercise.

2.3 Countability

As we recalled the notion of a set and certain related concepts. Subsequently, we discussed
certain properties of the sets of numbers N, Z, Q, R and C. A few more important properties and
related aspects concerning these sets are yet to be examined. One such significant aspect is the
countability of these sets. The concept of countability of sets was introduced by George Cantor
which forms a corner stone of Modern Mathematics.

2.3.1 Countable Sets

You can easily count the elements of a finite set. For example, you very frequently use the term
‘one hundred rupees’ or ‘fifty boxes’, ‘two dozen eggs’, etc. These figures pertain to the number
of elements of a set. Denote the number of elements in a finite set S by n (S). If S = {a, b, c, d}, then
n (S) = 4. Similarly n (S) = 26, if S is the set of the letters of English alphabet. Obviously, then
n () = 0, where  is the null set.
You can make another interesting observation when you count the number of elements of a
finite set. While you are counting these elements, you are indirectly and perhaps unconsciously,
using a very important concept of the one-one correspondence between two sets. Recall the
concept of one-one correspondence. Here one of the sets is a finite subset of the set of natural
numbers and the other set is the set consisting of the articles/objects like rupees, boxes, eggs, etc.
Suppose you have a basket of oranges. While counting the oranges, you are associating a natural
number to each of the oranges. This, as you know, is a one-one correspondence between the set
of oranges and a subset of natural members. Similarly, when you count the fingers of your
hands, you are in fact showing a one-one correspondence between the set of the fingers with a
subset, say N = (1, 2, .... 10) of N.
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Although, we have an intuitive idea of finite and infinite sets, yet we give a mathematical
definition of these sets in the following way:
Definition 7: Finite and Infinite Sets
A set S is said to be finite if it is empty or if there is a positive integer k such that there is one-one

correspondence between the elements of the set S and the set N ={1, 2, 3 ..... k}. A set is said to be
K
infinite if it is not finite.
The advantage of using the concept of one-one correspondence is that it helps in studying the
countability of infinite sets. Let E = {2, 4, 6, ....} be the set of even natural numbers. If we define a
mapping f: N  E as

f(n) = 2n " n  N,
then we find that f is a one-one correspondence between N and E.
Consider another examples, Suppose S = {1, 2, .... n} and T = {x , X , .... x }. Define a mapping
1 2 n
f: S T as
f(n) = x n  S.
n

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