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# 9.S: Finite and Infinite Sets (Summary)

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Important Definitions

• Equivalent sets, page 452
• Sets with the same cardinality, page 452
• Finite set, page 455
• Infinite set, page 455
• Cardinality of a finite set, page 455
• Cardinality of $$\mathbb{N}$$, page 466
• $$\aleph_0$$, page 466
• Countably infinite set, page 466
• Denumerable set, page 466
• Uncountable set, page 466

Important Theorems and Results about Finite and Infinite Sets

• Theorem 9.3. Any set equivalent to a finite nonempty set $$A$$ is a finite set and has the same cardinality as $$A$$.
• Theorem 9.6. If $$S$$ is a finite set and $$A$$ is a subset of $$S$$, then $$A$$ is finite and $$\text{card}(A) \le \text{card}(S)$$.
• Corollary 9.8. A finite set is not equivalent to any of its proper subsets.
• Theorem 9.9 [The Pigeonhole Principle]. Let $$A$$ and $$B$$ be finite sets. If $$\text{card}(A) > \text{card}(B)$$, then any function $$f: A \to B$$ is not an injection.
• Theorem 9.10. Let $$A$$ and $$B$$ be sets.

1. If $$A$$ is infinite and $$A \thickapprox B$$, then $$B$$ is infinite.
2. If $$A$$ is infinite and $$A \subseteq B$$,then $$B$$ is infinite.
• Theorem 9.13. The set $$\mathbb{Z}$$ of integers is countably infinite, and so card$$(\mathbb{Z}) = \aleph_0$$.
• Theorem 9.14. The set of positive rational numbers is countably infinite.
• Theorem 9.16. If $$A$$ is a countably infinite set and $$B$$ is a finite set, then $$A \cup B$$ is a countably infinite set.
• Theorem 9.17. If $$A$$ and $$B$$ are disjoint countably infinite sets, then $$A \cup B$$ is a countably infinite set.
• Theorem 9.18. The set $$\mathbb{Q}$$ of all rational numbers is countably infinite.
• Theorem 9.19. Every subset of the natural numbers is countable.
• Corollary 9.20. Every subset of a countable set is countable.
• Theorem 9.22. The open interval (0, 1) is an uncountable set.
• Theorem 9.24. Let $$a$$ and $$b$$ be real numbers with $$a < b$$. The open interval $$(a, b)$$ is uncountable and has cardinality $$c$$.
• Theorem 9.26. The set of real numbers $$\mathbb{R}$$ is uncountable and has cardinality $$c$$.
• Theorem 9.27 [Cantor’s Theorem]. For every set $$A$$, $$A$$ and $$\mathcal{P}(A)$$ do not have the same cardinality.
• Corollary 9.28. $$\mathcal{P}(\mathbb{N})$$ is an infinite set that is not countably infinite.
• Theorem 9.29 [Cantor-Schr$$\ddot{0}$$der-Bernstein]. Let $$A$$ and $$B$$ be sets. If there exist injections $$f_1: A \to B$$ and $$f_2: B \to A$$, then $$A \thickapprox B$$.

9.S: Finite and Infinite Sets (Summary) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Ted Sundstrom (ScholarWorks @Grand Valley State University) via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.