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This course will roughly cover the following topics and specific applications in computer science 1 Sets, functions and relations 2 Proof techniques and induction
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A Course in Discrete Structures
Rafael Pass
Wei-Lung Dustin Tseng
Preface
Discrete mathematics deals with objects that come indiscretebundles, e.g.,
1 or 2 babies. In contrast, continuous mathematics deals with objects that
varycontinuously, e.g., 3.42 inches from a wall. Think of digital watches versus analog watches (ones where the second hand loops around continuously without stopping). Why study discrete mathematics in computer science? It does not directly help us write programs. At the same time, it is the mathematics underlying almost all of computer science. Here are a few examples: Designing high-speed networks and message routing paths.
Finding good algorithms for sorting.
Performing web searches.
Analysing algorithms for correctness and eciency.
Formalizing security requirements.
Designing cryptographic protocols.
Discrete mathematics uses a range of techniques, some of which is sel- dom found in its continuous counterpart. This course will roughly cover the following topics and specic applications in computer science. 1.
Sets, functions an drelations
2.
Pro oftec hniquesand induction
3.
Num bertheory
a)
The math b ehindthe RSA Crypto sys tem
4.
Coun tingand com binatorics
5.
Probabilit y
a)
Spam detection
b)
F ormalsecurit y
6. Logic a)
Pro ofsof program correctness
7.
Graph theory
i a)Message Rou ting b)
So cialnet works
8.
Finite automata an dregular languages
a)
Compilers
In the end, we will learn to write precise mathematical statements that captures what we want in each application, and learn to prove things about these statements. For example, how will we formalize the infamous zero- knowledge property? How do we state, in mathematical terms, that a banking protocol allows a user to prove that she knows her password, without ever revealing the password itself?
Contents
Contents iii
1 Sets, Functions and Relations 1
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Set Cardinality, revisited . . . . . . . . . . . . . . . . . . . . . . 8
2 Proofs and Induction 13
2.1 Basic Proof Techniques . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Proof by Cases and Examples . . . . . . . . . . . . . . . . . . . 15
2.3 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Inductive Denitions . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Fun Tidbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Number Theory 37
3.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Modular Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 The EulerFunction . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Public-Key Cryptosystems and RSA . . . . . . . . . . . . . . . 56
4 Counting 61
4.1 The Product and Sum Rules . . . . . . . . . . . . . . . . . . . 61
4.2 Permutations and Combinations . . . . . . . . . . . . . . . . . 63
4.3 Combinatorial Identities . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Inclusion-Exclusion Principle . . . . . . . . . . . . . . . . . . . 69
4.5 Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . 72
5 Probability 73
iii
5.1 Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Conditional Probability and Independence . . . . . . . . . . . . 77
5.3 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Expectatation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.5 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6 Logic 95
6.1 Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Logical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 First Order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
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