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Introduction to Mathematical Proofs: A Transition

Charles E. Roberts, Jr.
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Textbooks in Mathematics
[Reviewed by
Miklós Bóna
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When writing a textbook whose main goal is to introduce students to the idea of proofs, an author faces many more choices than when writing, say, an introductory textbook on number theory or abstract algebra. A fair part of these choices are a matter of taste, and so two readers may have very different opinions of the same book.

I find the introductory part of this book too long. The book has a total of nine chapters, the first five of which, totaling more than 250 pages, can be called introductory. The chapters are about Logic, Formal Mathematical Systems and Proofs, Set Theory, Relations, and Functions. These chapters very slowly introduce the topics mentioned above. My problem is that the proofs that are used to teach what a proof is are not that interesting. Students will not get excited by a formal proof of a statement like “the square of n is even if and only if n is even”. Most students already know that, even if they cannot formally prove it. I know that formal proofs are the point, but keeping the student’s attention and enthusiasm is also the point. In my experience, it is better to prove something that is actually new and interesting for the student, and that way the student will learn that formal proofs are worthy and useful constructs.

Apart from a few examples, the book only reaches the point of interesting examples with chapter 6 (Induction) and chapter 7 (Cardinalities of Sets). These chapters have the right level of examples and exercises. Chapter 8 is on theorems from real analysis. Instead of this, present reviewer would have liked a chapter on complex numbers, since most students have learned some real analysis in calculus, but this might be their only chance of seeing complex numbers. Finally, the last chapter is on proofs from group theory. Given that only one chapter is given to this huge topic, the author only discusses the cyclic groups in detail, and has not much time to discuss non-abelian groups such as permutation groups or dihedral groups. The general linear groups are mentioned, though.

Finally, the book could have benefitted from a more thorough editing. On page 35, for example, in a boxed definition we read “a invalid argument” twice. Sentences ending in mathematical formulae often do not end with a period in this book, which is occasionally confusing.

Miklós Bóna is Professor of Mathematics at the University of Florida.


Statements, Negation, and Compound Statements

Truth Tables and Logical Equivalences

Conditional and Biconditional Statements

Logical Arguments

Open Statements and Quantifiers

Deductive Mathematical Systems and Proofs

Deductive Mathematical Systems

Mathematical Proofs

Set Theory

Sets and Subsets

Set Operations

Additional Set Operations

Generalized Set Union and Intersection



The Order Relations <, =, >, =

Reflexive, Symmetric, Transitive, and Equivalence Relations

Equivalence Relations, Equivalence Classes, and Partitions



Onto Functions, One-to-One Functions, and One-to-One Correspondences

Inverse of a Function

Images and Inverse Images of Sets

Mathematical Induction

Mathematical Induction

The Well-Ordering Principle and the Fundamental Theorem of Arithmetic

Cardinalities of Sets

Finite Sets

Denumerable and Countable Sets

Uncountable Sets

Proofs from Real Analysis


Limit Theorems for Sequences

Monotone Sequences and Subsequences

Cauchy Sequences

Proofs from Group Theory

Binary Operations and Algebraic Structures


Subgroups and Cyclic Groups

Appendix: Reading and Writing Mathematical Proofs

Answers to Selected Exercises