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Class 21: Reviewing Infinities cs2102: Discrete Mathematics uvacs2102.github.io 0 David Evans, Mohammad Mahmoody University of Virginia

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Reminder • Problem set 8 due this Friday • 2nd exam next Thursday. (We will post last years’ exam this week.) 1

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Today’s Plan • Reviewing: Infinite, Countable, and Uncountable Sets • More examples • Thursday: Number Thoery 2

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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 .

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Equal Cardinalities 4 If there exists a bijection between sets and , they have the same cardinality: || = ||

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Comparing Cardinalities 5 If there exists a surjective function from sets to , then we say ≤ || Equivalent: total injective function from to .

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6 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 4th Definition. A set is 4th def-infinite, if there is a total injective function from ℕ to

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Examples 7

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Countable 8 Definition. A set is countable if and only if ≤ |ℕ| Namely, there is a surjective function from ℕ to . Equivalently:

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Example: A,B are countable How about ∪ ? 9

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Example set of all finite subsets of N 10

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Countably Infinite 11 A set is countable if and only if there exists a surjective function from ℕ to . Namely ≤ |ℕ| A set is infinite, if and only if there is a surjective function from to ℕ. Namely ℕ ≤ || A set is countably infinite iff it is countable and it is infinite. Equivalent: a set is countably infinite iff there exists a bijection between and ℕ

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Is there anything bigger than countable? 12

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Cantor’s Theorem 13 For all sets , > | |.

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ℕ = ∈ ℝ, 0 ≤ < 1 | 14

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Charge Infinities are Spooky, Strange, and Surprising, We will mostly deal with countable infinities PS8 due this Friday Exam on Thurs next week 15