دانلود کتاب Theoremus. A Student’s Guide to Mathematical Proofs

فهرست مطالب :

Preface
Reference
Acknowledgements
Contents
Acronyms
Part I The Basics
1 Introduction
1.1 The World Has Gone Maths
1.2 The Culture and Tradition of Proofs
1.3 Proofs Sharpen Thinking Skills
References
2 Theorems and Proofs
2.1 What Are Theorems?
2.2 What Is an Argument?
2.3 What Is a Proof?
2.3.1 Fallacious Proofs
2.3.2 Valid and Sound
2.3.3 Notation Plays a Part
2.4 Reflections
3 Types of Theorems
3.1 What Are Theorems?
3.2 Statements and Propositions
3.3 If-Then
3.4 If and only If
3.5 Equational Statements
3.6 Quantified, ``for all\'\' Statements
3.7 Quantified, ``there exist\'\' Statements
3.8 Reflections
References
4 Logical Foundations of Proof
4.1 Propositional Logic
4.1.1 Basic Components
4.1.2 True/False Valuations
4.1.3 Logical Equivalences
4.1.4 Inference Rules
4.1.5 Reflections
4.2 Predicate Logic
4.2.1 Basic Components
4.2.2 Categorical Forms
4.2.3 True/False Valuations
4.2.4 Logical Equivalences
4.2.5 Inference Rules
4.3 Fallacy Alert
References
5 Types of Proof Techniques
5.1 Direct Method
5.2 Indirect Method
5.3 Proof by Contradiction
5.4 Left-Right Method
5.5 The Case Method
5.6 Mathematical Induction Method
5.6.1 Weak Form
5.6.2 Strong Form
5.6.3 Why It Works
5.7 Reflections
5.8 Proof by Construction
Reference
Part II An Application
6 Formal System for PL
6.1 PL as a Formal System
6.2 PL Syntax
6.3 Induction on PL Formula
6.4 PL Semantics
6.5 Satisfiability, Validity and Consequences
6.6 PL Proof System
6.6.1 Gentzen Style ND
6.6.2 Gentzen Inference Rules
6.7 Consistency, Completeness and Soundness
6.7.1 Particular
6.7.2 General
6.8 Resolution
6.8.1 Satisfiability and Consistency Again
6.8.2 The Normal Forms
6.8.3 The Method
References
7 Formal System for FOL
7.1 FOL as a Formal System
7.2 FOL Syntax
7.2.1 Terms
7.2.2 Formulas
7.2.3 Agreement
7.2.4 Substitution
7.3 FOL Semantics
7.4 FOL Proof System
7.4.1 Consistency, Completeness and Soundness
7.5 Resolution
7.5.1 Rectified Form
7.5.2 Prenex Form
7.5.3 Some Helpful Equivalences
7.5.4 Skolemizing
7.5.5 Unification
7.5.6 The Procedure
References
8 Doing the Math
8.1 First-Order Theories
8.1.1 Definition of Theories
8.1.2 Some Examples
8.2 The Production of Theorems
8.2.1 Crank Proofs
8.2.2 Real Serious Proofs
References
Index