lecture26-cardinality.pdf - CSE 311 Foundations of Computing Lecture 26 Cardinality Uncomputability Course Evaluation Online • Fill this out by Sunday

lecture26-cardinality.pdf - CSE 311 Foundations of...

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CSE 311: Foundations of Computing Lecture 26: Cardinality, Uncomputability

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Last time: Languages and Representations All Context-Free Regular Finite 0* DFA NFA Regex {001, 10, 12} e.g. palindromes, balanced parens, {0 n 1 n :n 0} ?

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General Computation
Computers from Thought Computers as we know them grew out of a desire to avoid bugs in mathematical reasoning. Hilbert in a famous speech at the International Congress of Mathematicians in 1900 set out the goal to mechanize all of mathematics . In the 1930s, work of Gödel and Turing showed that Hilbert’s program is impossible . Gödel’s Incompleteness Theorem Undecidability of the Halting Problem Both of these employ an idea we will see called diagonalization . The ideas are simple but so revolutionary that their inventor Georg Cantor was shunned by the mathematical leaders of the time: Poincaré referred to them as a “grave disease infecting mathematics.” Kronecker fought to keep Cantor’s papers out of his journals. Cantor spent the last 30 years of his life battling depression, living often in “sanatoriums” (psychiatric hospitals).

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Cardinality What does it mean that two sets have the same size?
Cardinality What does it mean that two sets have the same size?

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1-1 and onto A function ࠵? ∶ ࠵? → ࠵? is one-to-one (1-1 ) if every output corresponds to at most one input; i.e. ࠵? ࠵? = ࠵? ࠵? ( ⇒ ࠵? = ࠵?′ for all ࠵?, ࠵? ( ∈ ࠵?. A function ࠵? ∶ ࠵? → ࠵? is onto if every output gets hit; i.e. for every ࠵? ∈ ࠵? , there exists ࠵? ∈ ࠵? such that ࠵? ࠵? = ࠵? . a b c d e 1 2 3 4 5 6 1-1 but not onto ࠵? ࠵?
Cardinality Definition: Two sets ࠵? and ࠵? have the same cardinality if there is a one-to-one correspondence between the elements of ࠵? and those of ࠵? . More precisely, if there is a 1-1 and onto function ࠵? ∶ ࠵? → ࠵? . ࠵? ࠵? a b c d e 1 2 3 4 5 6 f The definition also makes sense for infinite sets!

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Cardinality Do the natural numbers and the even natural numbers have the same cardinality? Yes! 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ... 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 ... What’s the map ࠵? ∶ ℕ → ࠵?ℕ ? ࠵? ࠵? = ࠵?࠵?
Countable sets Definition : A set is countable iff it has the same cardinality as some subset of .

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