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Passage to Abstract Mathematics

Mark E. Watkins and Jeffrey L. Meyer
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
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Here’s a quote from the preface of a book that was published in 1967:

At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After calculus, he takes a course in analysis and a course in algebra. Depending on his interests (or those of his department), he takes courses in special topics…The exciting revelations that that there is some unity in mathematics, that fields overlap, that techniques of one field have applications in another, are denied the undergraduate.

(Lecture Notes on Elementary Topology and Geometry, by I.M. Singer and John A. Thorpe)

And the following passage is contained in the preface of the book under review:

Too often in an undergraduate mathematics curriculum, each course or sequence is taught as a discrete entity, as though the intersection of its content with the content of other courses or sequence were empty.

So, after a period of 44 years, it seems that the problem of the compartmentalized mathematics curriculum is still a cause of concern for at least a few of those involved in the teaching of the subject. The authors of this book clearly belong to this minority.

But the contents of this book are very different to that written by Singer and Thorpe — although the underlying philosophy is the same. Here, the foundations are provided for the later study of topics such as algebra, analysis and topology, and readers are simultaneously led to see mathematics as a way of knowing. In other words, there is a good balance between process and content.

The content is centred upon logic, proof, sets, relations and functions. Methods of proof are explored, and deeper ideas such as cardinality and infinity introduced. The book concludes with a short chapter called ‘Algebraic Systems’, with the notion of binary operation being introduced by means of examples previously discussed. These include logical connectives, set operations, function composition and the number operations. The text is nicely balanced between expository narrative, examples, illustrations and exercises, and the book as a whole is not overloaded with content. Consequently, it makes for pleasurable and informative reading.

In general, I think that this book provides a very useful first step into ‘abstract mathematics’, because many essential concepts, and the vocabulary and notation that represent them (e.g., quantifiers), are systematically developed. The exercises are of varying degrees of difficulty, but the authors have deliberately provided no solutions or hints.

In conclusion, I would like to make two minor observations regarding the contents. Firstly, as with many books that introduce symbolic logic, I could find no explicit discussion of the validity of an argument. That is to say, an argument is valid if the conjunction of the premises implies the conclusion, which is a very useful framework in which to embed ideas of mathematical proof. Secondly, the concepts of binary operation and algebraic systems require a wider range of examples than is possible to provide in a book of this size. However, when students subsequently encounter topics like geometric transformations, matrices and vector spaces etc, they will have a much wider range of examples to draw upon.

Overall, I feel that this book is highly suited to the purpose referred to in its title.

As an undergraduate student in 1962, Peter Ruane greatly benefitted from a book with a similar aim to this one. It was Finite Mathematical Structures, by Mirkil, Kemeny, Snell and Thompson — possibly the first ever text written for foundation course.


Preface for the Instructor vii

Preface for the Student xiii


1. Logic and Proof

1.1 Proofs, what and why?

1.2 Statements and Non-statements

1.3 Logical Operations and Logical Equivalence

1.4 Conditionals, Tautologies and Contradictions

1.5 Methods of Proof

1.6 Quantifiers

1.7 Further Exercises


2. Numbers

2.1 Basic Ideas of Sets

2.2 Sets of Numbers

2.3 Some properties of N and Z

2.4 Prime Numbers

2.5 gcd’s and lcm’s

2.6 Euclid’s Algorithm

2.7 Rational Numbers and Algebraic Numbers

2.8 Further Exercises


3. Sets

3.1 Subsets

3.2 Operations with Sets

3.3 The Complement of a Set

3.4 The Cartesian Product

3.5 Families of Sets

3.6 Further Exercises


4. Induction

4.1 An Inductive Example

4.2 The Principle of Mathematical Induction

4.3 The Principle of Strong Induction

4.4 The Binomial Theorem

4.5 Further Exercises


5. Functions

5.1 Functional Notation

5.2 Operations on Functions

5.3 Induced Set Functions

5.4 Surjections, Injections, and Bijections

5.5 Identity Functions, Cancellation, Inverse Functions, and Restrictions

5.6 Further Exercises


6. Binary Relations

6.1 Partitions

6.2 Equivalence Relations

6.3 Order Relations

6.4 Bounds and Extremal Elements

6.5 Applications to Calculus

6.6 Functions Revisited

6.7 Further Exercises


7. Infinite Sets and Cardinality

7.1 Counting

7.2 Properties of Countable Sets

7.3 Counting Countable Sets

7.4 Binary Relations on Cardinal Numbers

7.5 Uncountable Sets

7.6 Further Exercises


8. Algebraic Systems

8.1 Binary Operations

8.2 Modular Arithmetic

8.3 Numbers Revisited

8.4 Complex Numbers

8.5 Further Exercises


Index of Symbols and Notation