Class 17

Countable Infinity

Transcript

1. Class 17: Infinite Sets cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans Mohammad Mahmoody University of Virginia
2. Plan for today – Defining Infinite Sets – How to compare size/cardinality of infinite sets. – Today: countable sets – [Thursday: uncountable sets]
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. Infinite Sets 3 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. Infinities before 1800’s 4
6. Will Class Ever End? 5 1 2 + 1 4 + 1 8 + ⋯ =
7. Zeno’s Paradox 6 0 50m 100m
8. 7 Newton’s Differential Calculus (1671, published 1742) Leibniz’s Calculus (1684)
9. 8 Georg Cantor (1845-1918) “corruptor of youth” Leopold Kronecker “utter nonsense” Ludwig Wittgenstein “grave disease” Henri Poincaré
10. Cardinality of Finite Sets 9 The cardinality of the set ℕ = ∈ ℕ ∧ < } is . If there is a bijection between two sets, they have the same cardinality. from Class 9:
11. Infinite Sets 10 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 .
13. How to talk about size of Infinite Sets? 12
14. Same Cardinality of Infinite Sets 13 If there exists a bijection between sets and , they have the same cardinality: || = ||
16. Also we can compare Cardinality of Infinite Sets 15 If there exists a surjective function from sets to , then we say ≤ ||
17. Two Useful Facts 1. || = || implies || ≤ || and ≤ || 2. If ≤ || and || ≤ || then = || proof? Not easy.. Schroder-Bernstein theorem.. 16
18. Dedekind’s Definition of Infinite Sets 17 Definition. A set is Dedekind-infinite if and only if it has the same cardinality as some strict subset of itself. Standard Def. A set is infinite, if there is no bijection between and any ℕ .
19. 18 Definition. A set is Dedekind-infinite if and only if it has the same cardinality as some strict subset of itself. Is ℕ Dedekind-infinite?
20. 19 Definition. A set is Dedekind-infinite if and only if it has the same cardinality as some strict subset of itself. Is ℕ Dedekind-infinite for any arbitrary large ?
21. Definitions of Infinity 20 Definition. A set is Dedekind-infinite if and only if it has the same cardinality as some strict subset of itself. Standard Def. A set is infinite, if there is no bijection between and any ℕ .
22. 21 Definition. A set is Dedekind-infinite if and only if it has the same cardinality as some strict subset of itself. Standard Def. A set is infinite, if there is no bijection between and any ℕ . Are these definitions equivalent?
23. 22 1st : Definition. A set is Dedekind-infinite if and only if it has the same cardinality as some strict subset of itself. 2nd : Standard Definition. A set is infinite, if there is no bijection between and any ℕ .
24. 23 1st : Definition. A set is Dedekind-infinite if and only if it has the same cardinality as some strict subset of itself. 2nd : Standard Definition. A set is infinite, if there is no bijection between and any ℕ .
26. 25 1st : Definition. A set is Dedekind-infinite if and only if it has the same cardinality as some strict subset of itself. 2nd : Standard Definition. A set is infinite, if there is no bijection between and any ℕ . 3rd Definition. A set is 3rd def-infinite, if ℕ ≤ ||. Namely, there is a surjective function from to ℕ
27. Generalizing counting to infinite sets.. 26
28. Countable 27 Definition. A set is countable if and only if ≤ |ℕ| Namely, there is a surjective function from ℕ to .
29. Countably Infinite 28 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.
30. Countably Infinite 29 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 iff it is countable and it is infinite. Equivalent definition: A set is countably infinite iff there exists a bijection between and ℕ….. (prove this using Schroder-Bernstein theorem)