1549901369-Elements_of_Real_Analysis__Denlinger_

2.8 *countable and Uncountable Sets 129

y =f. Xm, since y and Xm differ in the mth decimal place and em =f. 0, 9;

Thus, we have constructed a real number y E (0, 1), which in not in the

list (16). Contradiction! Therefore, it is impossible to list the elements of (0, 1)

as a sequence. •

The following is a very surprising result, which we can now prove. It says

that the irrational numbers are more numerous than the rational numbers. This

may come as a surprise, if you have always thought of irrational numbers as

exceptional cases.

Corollary 2.8.8 The set of irrational numbers is uncountable (hence, more

numerous than the set of rationals).

Proof. Exercise 2. •

Significance of Theorems 2.8.5 and 2.8. 7 and Their Corollaries:
Taken together, these results give us profound insight into a fundamental con-
trast between the set of rational numbers and the sets of real numbers and
irrational numbers. The set Q can be put into 1-1 correspondence with N,
whereas JR and the set of irrational numbers cannot. The set of rational num-
bers is countable, whereas JR and the set of irrational numbers are not. Said
another way, the rational numbers are not more numerous than the
natural numbers, but the irrational numbers and the real numbers
are.
further fascinating results of Cantor on infinite sets may be found in Sec-
tion 3.4.

EXERCISE SET 2.8

1. Prove that the union of two denumerable sets is denumerable.


2. Prove Corollary 2.8.8. [Hint: Use Exercise l.]


3. Prove that the union of a denumerable collection of denumerable sets is
   denumerable. [Hint: Use a diagonal procedure like the one used in proving
   Theorem 2.8.5. Double subscripts may help.]


4. Prove that if A and B are denumerable sets, then so is A x B. [In fact, if
   A 1 , A 2 , · · · , An are denumerable, then so is A 1 x A2 x · · · x An.]