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Class 18: Infinite Sets

David Evans
November 03, 2016
4.8k

Class 18: Infinite Sets

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

See course site for notes:
https://uvacs2102.github.io

David Evans

November 03, 2016
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Transcript

  1. Plan Infinite Cardinality Countable Sets Uncountable Sets 1 Exam 2

    – see notes! Dave’s office hours cancelled Wednesday PS8 Due Friday Finish reading Ch 8 before Thursday
  2. Cardinality of Finite Sets 2 The cardinality of the set

    " = ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality. from Class 9:
  3. Infinite Sets 4 The cardinality of the set ℕ" =

    ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality.
  4. Infinite Sets 5 The cardinality of the set ℕ" =

    ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality. A set is infinite, if there is no bijection between and any ℕ" .
  5. 11 1 2 + 1 4 + 1 8 +

    ⋯ If … means ∀ ∈ ℕ terms elided: If … means actually infinitely many terms elided: 1 2 + 1 4 + 1 8 + ⋯
  6. 18 Georg Cantor (1845-1918) “corruptor of youth” Leopold Kronecker “utter

    nonsense” Ludwig Wittgenstein “grave disease” Henri Poincaré
  7. Same Cardinality: Equipollent 19 If there exists a bijection between

    sets and , they have the same cardinality: || = ||
  8. 20

  9. 21

  10. Dedekine’s Definition 22 Definition. A set is Dedekine-infinite if and

    only if it has the same cardinality as some strict subset of itself.
  11. 23 Definition. A set is Dedekine-infinite if and only if

    it has the same cardinality as some strict subset of itself. Is ℕ Dedekine-infinite?
  12. 24 Definition. A set is Dedekine-infinite if and only if

    it has the same cardinality as some strict subset of itself. Is ℕ"Dedekine-infinite for all ?
  13. Definitions of Infinity 25 Definition. A set is Dedekine-infinite if

    and only if it has the same cardinality as some strict subset of itself. Standard Definition. A set is infinite, if there is no bijection between and any ℕ" .
  14. 26 Definition. A set is Dedekine-infinite if and only if

    it has the same cardinality as some strict subset of itself. Standard Definition. A set is infinite, if there is no bijection between and any ℕ" . Are these definitions equivalent?
  15. 27

  16. 28

  17. Countable 29 Definition. A set is countable if and only

    if there exists a surjective function from ℕ to .
  18. 30 A set is countable if and only if there

    exists a surjective function from ℕ to . Prove all finite sets are countable. A set is infinite, if there is no bijection between and any ℕ" .
  19. 31 A set is countable if and only if there

    exists a surjective function from ℕ to . Prove ℤ is countable. A set is infinite, if there is no bijection between and any ℕ" .
  20. Countably Infinite 32 A set is countable if and only

    if there exists a surjective function from ℕ to . A set is infinite, if there is no bijection between and any ℕ" . A set is countably infinite if it is countable and it is infinite.
  21. Countably Infinite 33 A set is countable if and only

    if there exists a surjective function from ℕ to . A set is infinite, if there is no bijection between and any ℕ" . A set is countably infinite if it is countable and it is infinite. A set is countably infinite if there exists a bijection between and ℕ.
  22. Charge Thursday: sets that are not countable! (This is what

    got people really upset with Cantor) Read MCS Chapter 8 PS 8 due Friday Exam 2 on Nov 10 37