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Mathematics LibreTexts

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 N, page 466
  • 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 card(A)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 card(A)>card(B), then any function f:AB is not an injection.
  • Theorem 9.10. Let A and B be sets.

    1. If A is infinite and AB, then B is infinite.
    2. If A is infinite and AB,then B is infinite.
  • Theorem 9.13. The set Z of integers is countably infinite, and so card(Z)=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 AB is a countably infinite set.
  • Theorem 9.17. If A and B are disjoint countably infinite sets, then AB is a countably infinite set.
  • Theorem 9.18. The set 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 R is uncountable and has cardinality c.
  • Theorem 9.27 [Cantor’s Theorem]. For every set A, A and P(A) do not have the same cardinality.
  • Corollary 9.28. P(N) is an infinite set that is not countably infinite.
  • Theorem 9.29 [Cantor-Schr¨0der-Bernstein]. Let A and B be sets. If there exist injections f1:AB and f2:BA, then AB.

This page titled 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 the style and standards of the LibreTexts platform.

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