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Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njb2b/theory Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death

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Today: Formal Logic Review of discrete math Notation will be used heavily Unambiguous All answers require argument Goals: Develop intuition Practice fluency Note: today we’ll be more formal than I expect on the problems

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Propositional Logic

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First Order Logic

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First Order Logic Inference

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First Order Logic

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General Proof Guidelines No need to be super formal Shorter is always better! “A picture is worth 1000 words” Don’t try to surprise people Consider all corner cases

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Set Theory

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Notation: {}: Set notation {1, 2, 3} is a set {1, 2, 3, …} is the set of all positive whole numbers

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Set Theory

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Useful Sets

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Problem: (1/4) + (1/4) 2 + (1/4) 3 + (1/4) 4 + … = ? Find a short, geometric, induction-free proof. 1 1

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Problem: (1/8) + (1/8) 2 + (1/8) 3 + (1/8) 4 + …= ? Find a short, geometric, induction-free proof.

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Binary Relations

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Relationships Between Structures “→” ≝ “Can be defined in terms of” Programs Groups Proofs Trees Complex numbers Operators Propositions Graphs Real.

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