The preface of this book greets the reader with the statement ‘Welcome to the study of mathematical reasoning’, surely an indictment of the way mathematics is taught at all stages up to the high school level. High school students may have seen a few things proved here and there — but they will have been taught nothing about proof in general. For example, the word ‘Pythagoras’ too often denotes a rule about triangles, whose justification is that it seems to work.
Generally speaking, schools do not even encourage students to develop their own informal methods of justification. An example of this is given by part (a) of exercise 11 in section 1.4 (Basic Methods of Proof). Here, the reader is asked to decide whether the following statement is a proof of the fact that, if a is an odd integer, then a2 + 1 is even:
‘Let a be odd. Then, by squaring an odd, we get an odd. An odd plus an odd is even. So a2 + 1 is even’
Had this statement been made by a student at the primary or secondary school level, it would be a good indication that he/she had already been introduced to the ‘study of mathematical reasoning’. Unfortunately, without such informal preparation for encounter with formal mathematical proof, students will find it all the more difficult when they eventually meet it as undergraduates.
Anyway, one of the aims of this book, now in its 7th edition, is to introduce the reader to the main types of mathematical proof, and to analyse the logical basis of each one. The other main aim is to develop ‘major ideas’ needed for continuing work’. Such ideas are dealt with in the first five of the book’s chapters, which are called ‘Logic and Proof’, ‘Set Theory’, ‘Relations and Partitions’, ‘Functions’ and ‘Cardinality’. Examples of the advanced mathematics, to which the title refers, appear throughout the book, but particularly in the last two chapters: ‘Concepts of Algebra’ and ‘Concepts of Analysis’.
Written in classic textbook style, the book introduces each topic by means of very basic examples. Hundreds of carefully graded exercises ease the path to the rigorous treatment of more difficult ideas (e.g., the Axiom of Choice, the Heine-Borel theorem etc). The all-pervading theme is that of proof and that many different forms of proof appear in a variety of mathematical contexts. Many of the exercises require students to grade various ‘attempts’ at proofs, which is one of the book’s many strong points. Solutions or hints are provided for a good proportion of the included exercises.
This latest edition is based upon the same goals and core materials as previous editions, but with some re-organization and many new examples and exercises. Pre-requisite knowledge is now more clearly defined and full details of the changes are provided in the preface. I have no doubt in saying that this book would serve as excellent basis for a foundation course in any undergraduate mathematics programme.
Peter Ruane has retired from the very pleasant task of preparing students to meet the challenge of teaching mathematics in primary and secondary schools in the UK.
1. LOGIC AND PROOFS.
Propositions and Connectives. Conditionals and Biconditionals. Quantifiers. Basic Proof Methods I. Basic Proof Methods II. Proofs Involving Quantifiers. Additional Examples of Proofs
2. SET THEORY.
Basic Notions of Set Theory. Set Operations. Extended Set Operations and Indexed Families of Sets. Induction. Equivalent Forms of Induction. Principles of Counting.
3. RELATIONS AND PARTITIONS.
Relations. Equivalence Relations. Partitions. Ordering Relations. Graphs.
Functions as Relations. Constructions of Functions. Functions That Are Onto; One-to-One Functions. One-to-One Correspondences and Inverse Functions. Images of Sets. Sequences.
Equivalent Sets; Finite Sets. Infinite Sets. Countable Sets. The Ordering of Cardinal Numbers. Comparability of Cardinal Numbers and the Axiom of Choice.
6. CONCEPTS OF ALGEBRA: GROUPS.
Algebraic Structures. Groups. Subgroups. Operation Preserving Maps. Rings and Fields.
7. CONCEPTS OF ANALYSIS: COMPLETENESS OF THE REAL NUMBERS.
Ordered Field Properties of the Real Numbers. The Heine-Borel Theorem. The Bolzano-Weierstrass Theorem. The Bounded Monotone Sequence Theorem. Comparability of Cardinals and the Axiom of Choice.