Set Theory: A First Course by Daniel W. Cunningham

Author:Daniel W. Cunningham
Language: eng
Format: epub, pdf
Publisher: Cambridge University Press
Published: 2016-06-21T04:00:00+00:00

Corollary 5.2.7. A set is countably infinite if and only if there is a bijection .

Proof. If is countably infinite, then Theorem 5.2.6 implies that there exists a bijection . Conversely, suppose that is a bijection. Since is one-to-one, is infinite by Corollary 5.1.11. As is onto , Theorem 3.3.18 asserts the existence of the inverse function that is one-to-one. Hence, is countable and infinite. ☐

Theorem 5.2.8. If is a countably infinite set, then there exists an enumeration of all the elements in such that each element in appears in this enumeration exactly once.

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