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Notes on Discrete Mathematics

James Aspnes

2022-06-08 10:27

iCopyrightc?2004-2022 by James Aspnes. Distributed under a Creative Com- mons Attribution-ShareAlike 4.0 International license:https://creativecommons. org/licenses/by-sa/4.0/.

List of figures

xv ii

List of tables

xix

List of algorithms

Preface

xxi

Resources

xxii

1 Introduction

1.1 So why do I need to learn all this nasty mathematics?

1.2 But isn"t math hard?

1.3 Thinking about math with your heart

1.4 What you should know about math

1.4.1 Foundations and logic

1.4.2 Basic mathematics on the real numbers

1.4.3 Fundamental mathematical objects

1.4.4 Modular arithmetic and polynomials

1.4.5 Linear algebra

1.4.6 Graphs

1.4.7 Counting

1.4.8 Probability

1.4.9 Tools

2 Mathematical logic

2.1 The basic picture

2.1.1 Axioms, models, and inference rules

9 ii

CONTENTSiii

2.1.2 Consistency

2.1.3 What can go wrong

2.1.4 The language of logic

2.1.5 Standard axiom systems and models

2.2 Propositional logic

2.2.1 Operations on propositions

2.2.1.1 Precedence

2.2.2 Truth tables

2.2.3 Tautologies and logical equivalence

2.2.3.1 Inverses, converses, and contrapositives

2.2.3.2 Equivalences involving true and false

Example

2.2.4 Normal forms

2.3 Predicate logic

2.3.1 Variables and predicates

2.3.2 Quantifiers

2.3.2.1 Universal quantifier

2.3.2.2 Existential quantifier

2.3.2.3 Negation and quantifiers

2.3.2.4 Restricting the scope of a quantifier

2.3.2.5 Nested quantifiers

2.3.2.6 Examples

2.3.3 Functions

2.3.4 Equality

2.3.4.1 Uniqueness

2.3.5 Models

2.3.5.1 Examples

2.4 Proofs

2.4.1 Inference Rules

2.4.2 Proofs, implication, and natural deduction

2.4.2.1 The Deduction Theorem

2.4.2.2 Natural deduction

2.4.3 Inference rules for equality

2.4.4 Inference rules for quantified statements

2.5 Proof techniques

2.6 Examples of proofs

2.6.1 Axioms for even numbers

2.6.2 A theorem and its proof

2.6.3 A more general theorem

2.6.4 Something we can"t prove

CONTENTSiv

3 Set theory

3.1 Naive set theory

3.2 Operations on sets

3.3 Proving things about sets

3.4 Axiomatic set theory

3.5 Cartesian products, relations, and functions

3.5.1 Examples of functions

3.5.2 Sequences

3.5.3 Functions of more (or less) than one argument

3.5.4 Composition of functions

3.5.5 Functions with special properties

3.5.5.1 Surjections

3.5.5.2 Injections

3.5.5.3 Bijections

3.5.5.4 Bijections and counting

3.6 Constructing the universe

3.7 Sizes and arithmetic

3.7.1 Infinite sets

3.7.2 Countable sets

3.7.3 Uncountable sets

3.8 Further reading

4 The real numbers

4.1 Field axioms

4.1.1 Axioms for addition

4.1.2 Axioms for multiplication

4.1.3 Axioms relating multiplication and addition

4.1.4 Other algebras satisfying the field axioms

4.2 Order axioms

4.3 Least upper bounds

4.4 What"s missing: algebraic closure

4.5 Arithmetic

4.6 Connection between the reals and other standard algebras

4.7 Extracting information from reals

5 Induction and recursion

5.1 Simple induction

5.2 Alternative base cases

5.3 Recursive definitions work

5.4 Other ways to think about induction

CONTENTSv

5.5 Strong induction

5.5.1 Examples

5.6 Recursively-defined structures

5.6.1 Functions on recursive structures

5.6.2 Recursive definitions and induction

5.6.3 Structural induction

6 Summation notation

6.1 Summations

6.1.1 Formal definition

6.1.2 Scope

6.1.3 Summation identities

6.1.4 Choosing and replacing index variables

6.1.5 Sums over given index sets

97
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[PDF] [PDF] Notes on Discrete Mathematics - Computer Science