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cs103x-notes - Discrete Structures Lecture Notes Vladlen...

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Unformatted text preview: Discrete Structures Lecture Notes Vladlen Koltun 1 Winter 2007 1 Computer Science Department, 353 Serra Mall, Gates 464, Stanford University, Stanford, CA 94305, USA; vladlen@stanford.edu . Contents 1 Sets and Notation 1 1.1 Defining sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 More sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Induction 5 2.1 Introducing induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Strong induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Why is the induction principle true? . . . . . . . . . . . . . . . . . . . . . . 8 3 More Proof Techniques 11 3.1 Proofs by contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Direct proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Divisibility 15 4.1 The division algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Remainders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Greatest common divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 Greatest common divisors and linear combinations . . . . . . . . . . . . . . 18 5 Prime Numbers 21 5.1 The fundamental theorem of arithmetic . . . . . . . . . . . . . . . . . . . . 21 5.2 The infinity of primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6 Modular Arithmetic 25 6.1 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6.2 Modular division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 7 Relations and Functions 31 7.1 Ordered pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 7.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.3 Kinds of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.4 Creating relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8 Mathematical Logic 37 8.1 Propositions and predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.2 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 i 8.3 Negations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 8.4 Logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 8.5 Tautologies and logical inference . . . . . . . . . . . . . . . . . . . . . . . . 41 9 Counting 43 9.1 Fundamental principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 9.2 Basic counting problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 10 Binomial Coefficients 49 10.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10....
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