# doc1 - Notes on Discrete Mathematics Miguel A. Lerma...

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Notes on Discrete Mathematics Miguel A. Lerma

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Contents Introduction 5 Chapter 1. Logic, Proofs 6 1.1. Propositions 6 1.2. Predicates, Quantiﬁers 11 1.3. Proofs 13 Chapter 2. Sets, Functions, Relations 19 2.1. Set Theory 19 2.2. Functions 27 2.3. Relations 32 Chapter 3. Algorithms, Integers 38 3.1. Algorithms 38 3.2. The Euclidean Algorithm 48 3.3. Modular Arithmetic, RSA Algorithm 52 Chapter 4. Induction, Recurences 59 4.1. Sequences and Strings 59 4.2. Mathematical Induction 62 4.3. Recurrence Relations 65 Chapter 5. Counting 69 5.1. Basic Principles 69 5.2. Combinatorics 71 5.3. Generalized Permutations and Combinations 73 5.4. Binomial Coeﬃcients 75 5.5. The Pigeonhole Principle 77 Chapter 6. Probability 78 6.1. Probability 78 Chapter 7. Graph Theory 82 7.1. Graphs 82 7.2. Representations of Graphs 88 7.3. Paths and Circuits 91 3

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CONTENTS 4 7.4. Planar Graphs 97 Chapter 8. Trees 100 8.1. Trees 100 8.2. Binary Trees 102 8.3. Decision Trees, Tree Isomorphisms 104 8.4. Tree Transversal 113 8.5. Spanning Trees 116 Chapter 9. Boolean Algebras 122 9.1. Combinatorial Circuits 122 9.2. Boolean Functions, Applications 127 Chapter 10. Automata, Grammars and Languages 133 10.1. Finite State Machines 133 10.2. Languages and Grammars 137 10.3. Language Recognition 144 Appendix A. 150 A.1. Eﬃcient Computation of Powers Modulo m 150 A.2. Machines and Languages 152
Introduction These notes are intended to be a summary of the main ideas in course CS 310: Mathematical Foundations of Computer Science . I may keep working on this document as the course goes on, so these notes will not be completely ﬁnished until the end of the quarter. The textbook for this course is Keneth H. Rosen: Discrete Mathe- matics and Its Applications , Fifth Edition, 2003, McGraw-Hill. With few exceptions I will follow the notation in the book. These notes contain some questions and “exercises” intended to stimulate the reader who wants to play a somehow active role while studying the subject. They are not homework nor need to be addressed at all if the reader does not wish to. I will recommend exercises and give homework assignments separately. Finally, if you ﬁnd any typos or errors, or you have any suggestions, please, do not hesitate to let me know. Miguel A. Lerma mlerma@math.northwestern.edu Northwestern University Spring 2005 http://www.math.northwestern.edu/~mlerma/courses/cs310-05s/ 5

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CHAPTER 1 Logic, Proofs 1.1. Propositions A proposition is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “ x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its
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## This note was uploaded on 10/27/2008 for the course CS 8878 taught by Professor Mikejameson during the Spring '08 term at American Academy of Art.

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doc1 - Notes on Discrete Mathematics Miguel A. Lerma...

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