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Logic1028 - Infinity ZE 1 2 3 Subsets Sizes of sets Cantors...

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@ f E° °°° °° Infinity 1. Subsets 2. Sizes of sets 3. Cantor’s theorem
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Midterm graded Any questions? HW #6 (due 11/4) 1. Problem 13.12 (prove by a tableau) 2. Exercise 13.7 3. Show that the set of all rational num- bers (of the form a/b where a and b are integers – could be negative – and b≠0) is denumerable.
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Plan for the second half Cover Part III-V (Chs. 14-23) Also Ch.27 if there is time left over Gets more and more difficult as we move on to the later chapters.
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In chs.14-16, we will discuss concept of infinity (Ch.14) proof technique known as mathematical induction (Ch.15) These discussions will be useful for the remainder of the class. Esp. in the proofs of the correctness and completeness for propositional logic & first-order logic (chs.17-18)
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Today’s lecture About different sizes of sets, esp. of infin- ite sets Central result: Cantor’s theorem Implication: there are infinitely many differ- ent sizes among infinite sets! {04D990CC-6D54-4556-8FA0-77C3F685A9E5} N {79072FC5-4760-48C4-91A5-C3F68057BA7E} P(N) {3B7BE35F-00BB-4433-BF49-4A679EC69190} P(P(N)) {A87C6849-B7C5-42DF-8099-258922DD6356} P(P(P(N))) {08526670-59D4-416E-8ABE-5FDE93620D1E}
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A set is a collection of objects. Those objects are called members of the set. x is a member of a set A: x° A. x doesn’t belong A: xE A. A set A is a subset of a set B iff every member of A is also a member of B. In symbols, A°B.
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