[PDF] Notes on Discrete Mathematics - Computer Science PDF notes.pdf

31 déc 2020 · These are the notes for the Fall 2017 semester version of the Yale course CPSC 202a, Mathematical Tools for Computer Science They have

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[PDF] Notes on Discrete Mathematics - Computer Science

31 déc 2020 · These are the notes for the Fall 2017 semester version of the Yale course CPSC 202a, Mathematical Tools for Computer Science They have

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Discrete Mathematics for Computer Science Key College Publishing, Emeryville , Cali- fornia, 2006 Examinations There will be a final exam (covering the

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

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