An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

Citation

Eccles, P. J. (1997). An Introduction to Mathematical Reasoning: Numbers, Sets and Functions. Cambridge University Press.

Chapter Summary

After a preface, the book runs through six blocks of two to five short teaching chapters, each block closing with a chapter of problems (twenty-four teaching chapters in all). A long chapter of solutions to exercises and a bibliography end the volume. The summaries below follow the chapter titles.

The language of mathematics; Implications; Proofs; Proof by contradiction; The induction principle; Problems I

The first block introduces mathematical statements, implications and the standard forms of argument: direct proof, proof by contradiction and the induction principle. This is the part of the book that matches its stated purpose of teaching students to understand and construct proofs and to write clear mathematics.

The language of set theory; Quantifiers; Functions; Injections, surjections and bijections; Problems II

The second block turns to sets, quantifiers and functions, including the classification of functions into injections, surjections and bijections. These are the tools that every later block uses.

Counting; Properties of finite sets; Counting functions and subsets; Number systems; Counting infinite sets; Problems III

The third block is the combinatorial centre of the book. It begins with counting and finite sets, moves to counting functions and subsets, includes a chapter on the number systems, and ends by asking how infinite sets can be counted.

The division theorem; The Euclidean algorithm; Consequences of the Euclidean algorithm; Linear diophantine equations; Problems IV

The fourth block starts the number theory with the division theorem and the Euclidean algorithm, then follows their consequences through to linear diophantine equations.

Congruence of integers; Linear congruences; Congruence classes and the arithmetic of remainders; Partitions and equivalence relations; Problems V

The fifth block develops congruence of integers, the solution of linear congruences and the arithmetic of congruence classes, and closes with partitions and equivalence relations.

The sequence of prime numbers; Congruence modulo a prime; Problems VI

The last block treats the sequence of prime numbers and congruences modulo a prime.

Solutions to exercises

The solutions occupy about 46 pages, roughly an eighth of the book.

Key Concepts

The first two blocks: proof and sets

• Implication and proof: the logical form of "if … then" statements and the standard strategies for establishing them.
• Induction principle: the method for proving statements about all natural numbers.
• Quantifiers: the notation and meaning of "for all" and "there exists".
• Injection, surjection and bijection: the three properties of a function that the rest of the book uses to compare sets.

The third block: counting

• Counting functions and subsets: counting problems recast in terms of maps and subsets of finite sets.
• Counting infinite sets: the extension of counting to infinite sets.

The last three blocks: number theory

• Euclidean algorithm: the repeated-division method at the centre of the fourth block, with the consequences drawn from it.
• Linear diophantine equation: an equation in integers whose solvability is settled using that algorithm.
• Congruence classes and their arithmetic: arithmetic on remainders modulo n.
• Equivalence relation and partition: the general structure behind congruence classes.

Critical Analysis

Strengths

• A clear purpose. Cambridge describes the book as an introduction to the basic ideas of mathematical proof for students beginning university mathematics, with emphasis on understanding and constructing proofs and on writing clear mathematics.
• Proof learned through three subjects. Rather than treating logic in the abstract, the book practises proof on set theory, combinatorics and number theory, topics that Cambridge calls part of every mathematician's tool kit.
• Worked examples and class testing. The publisher lists a large number of fully worked examples and class testing by the author among the key features.
• A regular rhythm of practice. Every block ends with a problems chapter, and the solutions chapter is long enough to support self-study.
• A compact design. About 360 pages hold twenty-four teaching chapters of between 6 and 15 pages each, so a reader can treat each as a single study session.

Limitations

• A narrow range of topics. The chapter list is confined to logic and proof, sets and functions, counting, and elementary number theory. A student who also wants practice with proofs in analysis, linear algebra or algebra will need another book.
• Brief treatment of number systems and infinite sets. "Number systems" and "Counting infinite sets" are single chapters of about a dozen pages each, so readers who want a full construction of the number systems or a developed theory of cardinality will need a second book.

Where It Fits in the Course

Among the course's books on proof and logic, Eccles belongs with the proof-writing introductions rather than with the logic texts. Hammack's Book of Proof is a free introduction to proof methods that begins with sets, logic and counting, and Velleman's How to Prove It derives proof strategies from the logical form of a statement. Both put proof technique first. Eccles has a similar goal but spends nearly half the book on counting and elementary number theory, so it doubles as a first look at those subjects. Allenby's Numbers and Proofs is the closest relative in topic, since it also approaches proof through numbers and sets.

The logic-oriented books are a different kind of text. Leary and Kristiansen's A Friendly Introduction to Mathematical Logic and Enderton's A Mathematical Introduction to Logic study formal languages and deduction, which Eccles does not do. Liebeck's A Concise Introduction to Pure Mathematics surveys several areas of pure mathematics and works well as a next step after any of the proof books.

How to Use This Book

The book is meant for students starting university mathematics; school mathematics is the only prerequisite that the publisher indicates. It is also useful for anyone preparing for a first course in analysis or algebra who needs practice with proof.

Read the first two blocks in order, since they supply the language, the proof techniques and the function vocabulary used afterwards. A reader who has already worked through a proof text such as Hammack or Velleman can move more quickly through them. The third block (counting) and the number-theory blocks (fourth to sixth) can then be read in sequence.

Do the problems at the end of each block before moving on, and check solutions only after an honest attempt. Use the solutions chapter to see how a complete argument is written up, since writing clear mathematics is a stated aim of the book.