A compact and easily accessible book, it guides the reader in unravelling the apparent mysteries found in doing mathematical proofs. Simply written, it introduces the art and science of proving mathematical theorems and propositions and equips students with the skill required to tackle the task of proving mathematical assertions.
Theoremus - A Student's Guide to Mathematical Proofs is divided into two parts. Part 1 provides a grounding in the notion of mathematical assertions, arguments and fallacies and Part 2, presents lessons learned in action by applying them into the study of logic itself. The book supplies plenty of examples and figures, gives some historical background on personalities that gave rise to the topic and provides reflective problems to try and solve. The author aims to provide the reader with the confidence to take a deep dive into some more advanced work in mathematics or logic.
Author(s): Lito Perez Cruz
Edition: 1
Publisher: Springer
Year: 2021
Language: English
Commentary: Vector PDF
Pages: 138
City: New York, NY
Tags: First-Order Logic; Logic Fallacies; Discrete Mathematics; Proofs; Mathematical Induction; Propositional Logic
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
