Class 18: Infinite Sets

cs2102: Discrete Mathematics
University of Virginia, Fall 2016

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

Transcript

1. Class 18: Infinite Sets cs2102: Discrete Mathematics | F16 uvacs2102.github.io 0 David Evans University of Virginia
2. 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
3. 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:
4. Well-Ordering of Finite Sets 3 from Problem Set 5, Exam 1:
5. Infinite Sets 4 The cardinality of the set ℕ" = ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality.
6. 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" .
7. 6 http://cantorontheshore.blogspot.com/2014/10/the-real-meaning-of-to-infinity-and.html To Infinity and Beyond!
8. Will Class Ever End? 7 1 2 + 1 4 + 1 8 + ⋯ =
9. Zeno’s Paradox 8 0 50m 100m
10. 9 Raphael, School of Athens
11. 10 Zeno of Citium Wrong Zeno!
12. 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 + ⋯
13. Infinities before 1870s? 12
14. 13 Newton’s Differential Calculus (1671, published 1742) Leibniz’s Calculus (1684)
15. 14 Jefferson’s plan for the University of Virginia, Rockfish Gap Report 1818
16. 15 Jefferson’s plan for the University of Virginia, Rockfish Gap Report 1818
17. 16 Jefferson’s plan for the University of Virginia, Rockfish Gap Report 1818
18. 17 Georg Cantor (1845-1918)
19. 18 Georg Cantor (1845-1918) “corruptor of youth” Leopold Kronecker “utter nonsense” Ludwig Wittgenstein “grave disease” Henri Poincaré
20. Same Cardinality: Equipollent 19 If there exists a bijection between sets and , they have the same cardinality: || = ||
23. 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.
24. 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?
25. 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 ?
26. 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" .
27. 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?
30. Countable 29 Definition. A set is countable if and only if there exists a surjective function from ℕ to .
31. 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" .
32. 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" .
33. 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.
34. 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.
35. Example: ℤ is countably infinite 34
36. Example: (ℕ×ℕ) is countably infinite 35
37. Example: ℚ is countably infinite 36
38. 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