Page 42 - DMTH401_REAL ANALYSIS
P. 42
Real Analysis
Notes If S is denumerable and T is finite, then also we know that S T is denumerable. Hence S T is
countable. Again, if S is finite and T is denumerable, then again S T is denumerable and
countable.
Finally, if both S and T are denumerable, then S T is also denumerable and hence countable. In
fact, this result can be extended to countably many countable sets. We prove this in the following
theorem:
Theorem 3: The union of a countable number of countable sets is countable.
Proof: Let the given sets be A,, A , A,,.... Denote the elements of these sets, using double subscripts,
n
as follows:
A = {a , a , a , ....}
1 11 12 13
A = {a , a , a , ....}
2 21 22 23
A = {a , a , a , ....},
3 31 32 33
and so on. Note that the double subscripts have been used for the sake of convenience only. Thus
a is the jth element in the set A. Now, let us try to form a single list of all elements of the union
ij
of these given sets.
One method of doing this is by using Cantor’s diagonalised counting as indicated by arrows in
the following table:
Diagonalised Counting of A . j
=
i 1
Enlist the elements as indicated through the arrows. This is a scheme for making a single list of
all the elements.
Following the arrows in above table, you can easily arrive at the new single list:
a,,, a,,, a,,, a a > a , a,,, a,,, .......
31 22 13
Note that while doing so, you must omit the duplicates, if any.
Now, if any of the sets A,, A,, ......, are finite, then this will merely shorten the final list. Thus, we
have
A = {a,,, a , .....}, i = 1, 2, 3, ......
i i i i2
which each element appears only once. This set is countable and, so, complete the proof of the
theorem.
We are now in a position to discuss the countability of the sets of rational and real numbers.
2.3.2 Countability of Real Numbers
We have already established that the sets N and Z are countable. Let us, now, consider the case
of the set Q of rational numbers. For this we need the following theorems:
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