100% Mathematical Proof
Rowan Garnier (Richmond College London)
John Taylor (University of Brighton UK)
‘Proof’ has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. The exceptionally clear style and presentation ensures that the book will be useful and enjoyable to those studying and interested in the notion of mathematical ‘proof.’
Table of Contents
Chapter 1: Proofs: Mathematical and Non-mathematical
1.1 Introduction
1.2 Inductive and deductive reasoning
1.3 A proof or not a proof?
Chapter 2: Propositional Logic
2.1 Propositions and truth values
2.2 Logical connectives
2.3 Tautologies and contradictions
2.4 Logical implication and logical equivalence
2.5 Arguments and argument forms
2.6 Formal proof of the validity of arguments
2.7 The method of conditional proof
Chapter 3: Predicate Logic
3.1 Introduction
3.2 Quantification of propositional functions
3.3 Two-place predicates
3.4 Validation of arguments in predicate logic
Chapter 4: Axiom Systems and Formal Proof
4.1 Introduction
4.2 Case study of a proof
4.3 Axiom systems
4.4 Theorems and formal proofs
4.5 Informal proofs
Chapter 5: Direct Proof
5.1 The method of direct proof
5.2 Finding proofs
5.3 More advanced examples
Chapter 6: Direct Proofs: Variations
6.1 Introduction
6.2 Proof using the contrapositive
6.3 Proof by contradiction
6.4 Proof of a biconditional
Chapter 7: Existence and Uniqueness Proofs
7.1 Introduction
7.2 Proof by construction
7.3 Non-constructive existence proofs
7.4 Use of counter-examples
7.5 Uniqueness proofs
Chapter 8: Further Proof Techniques
8.1 Introduction
8.2 Proof of identities
8.3 Use of counting arguments
8.4 The method of exhaustion
Chapter 9: Mathematical Induction
9.1 The principle of mathematical induction
9.2 The second principle of mathematical induction
Appendix: Some definitions and terminology
References and Further Reading
Hints and solutions to selected exercises.
Index