Class 17: Infinite Sets cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans Mohammad Mahmoody University of Virginia 

Plan for today – Defining Infinite Sets – How to compare size/cardinality of infinite sets. – Today: countable sets – [Thursday: uncountable sets] 

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: 

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 . 

Infinities before 1800’s 4 

Will Class Ever End? 5 1 2 + 1 4 + 1 8 + ⋯ = 

Zeno’s Paradox 6 0 50m 100m 

7 Newton’s Differential Calculus (1671, published 1742) Leibniz’s Calculus (1684) 

8 Georg Cantor (1845-1918) “corruptor of youth” Leopold Kronecker “utter nonsense” Ludwig Wittgenstein “grave disease” Henri Poincaré 

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: 

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 . 

11 

How to talk about size of Infinite Sets? 12 

Same Cardinality of Infinite Sets 13 If there exists a bijection between sets and , they have the same cardinality: || = || 

14 

Also we can compare Cardinality of Infinite Sets 15 If there exists a surjective function from sets to , then we say ≤ || 

Two Useful Facts 1. || = || implies || ≤ || and ≤ || 2. If ≤ || and || ≤ || then = || proof? Not easy.. Schroder-Bernstein theorem.. 16 

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 . 

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? 

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 ? 

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 . 

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? 

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 . 

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 . 

24 

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 

Generalizing counting to infinite sets.. 26 

Countable 27 Definition. A set is countable if and only if ≤ |ℕ| Namely, there is a surjective function from ℕ to . 

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. 

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)