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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), .