# Section 2.8: Problem 6 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James . Munkres

(a) Show that
has cardinality at least
, where
is the set of rational numbers. Suggestion: Use Exercise 1.

(b) Show that
has cardinality at least
.

(a) According to Exercise 1, for every
, there is
such that
, but if
, then
, i.e.
, implying that
.

(b) There is a one-to-one function
, therefore, according to Exercise 2(a),
. In fact,
, as the sentence
, where
is the relation that defines
in
, is true in
, and, hence, in
. Moreover, since the sentence
is true in
, and, hence, in
,
is injective (one-to-one), and, according to (a),
.