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A Logical Introduction to Proof

Daniel W. Cunningham
Publication Date: 
Number of Pages: 
[Reviewed by
Annie Selden
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This is yet another textbook for a “transition-to-proofs” course. The last time I counted (in 2009) there were at least 25 such textbooks, some of which may now be out of print. Why another such book? Clearly, the author, who has taught the course using both Velleman’s How to Prove It: A Structured Approach and Epp’s Discrete Mathematics with Applications, thinks he has a new approach.

The major innovation seems to be his proof diagrams. In the preface, one finds the following example of a proof diagram for the proof of the statement \((\forall n\geq 1) P(n) \) by induction:

Prove \(P(1)\)

Let \(n\geq 1\)  be an integer.

    Assume \(P(n)\).

    Prove \(P(n+1)\).

“where the indentation is used to display a proof’s logical dependencies.” However, it is not clear to me that the inclusion of such proof diagrams goes beyond Velleman’s or Epp’s descriptions of structure in a useful way.

Something positive to note about this book is the author’s intent to have the book lead into abstract algebra and real analysis. Indeed, Chapters 8 and 9 deal with proving in these two content areas, instead of going on to proofs in number theory or graph theory, as some transition-to-proof course textbooks books do. There is also an appendix on proof strategies that may well be useful to students, as it puts them in an easily locatable part of the book. There is an index of special symbols in the back that may also be useful to students.

This book has nine chapters with way too many sections and subsections to cover in a semester. (I counted 101 subsections.) Indeed, the author suggests that a transition-to-proof course might just cover Chapters 1–7 (77 subsections), while omitting three subsections). Still 74 subsections seems a lot to cover in a semester.

The first two chapters are devoted to propositional and predicate logic (25 subsections). While such abstract coverage of logic seems to be standard in most transition-to-proof course textbooks, I feel this is way too much. Indeed, I prefer to have students attempt to construct proofs “from the get-go,” while providing logic “in a just in time” manner. This choice is substantiated by an examination by John Selden’s and my Ph.D. student, Milos Savic, of 42 student-constructed proofs from our transition-to-proof course. That study showed that very little logic beyond common sense (where modus ponens is considered common sense by us) is actually used in such proofs. That said, the author claims that the first two chapters can be covered quickly.

Chapter 3 covers proof strategies, and in my opinion is similar to Vellman’s Chapter 3 (at least in content). Chapter 4 is devoted to the Principle of Mathematical Induction (PMI), beginning with the Well Ordering Principle, which students are to use to prove the PMI.

Chapters 5 and 6 cover sets and functions, while Chapter 7 covers relations. This order is a bit unusual, as often relations are covered first and functions are then defined as a special kind of relation. There seem to be an adequate number of exercises at the end of each section. None of these are starred (*) as hard, and there are no answers or hints in the back of the book, both of which I consider a plus.

While there is an index in the back, it does not contain the phrase “proof diagram” — something I found a bit odd, as proof diagrams seem to be one of the claimed major innovations of this textbook.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education.

Preface.- The Greek Alphabet.- 1. Propositional Logic.- 2. Predicate Logic.- 3. Proof Strategies and Diagrams.- 4. Mathematical Induction.- 5. Set Theory.- 6. Functions.- 7. Relations.- 8. Core Concepts in Abstract Algebra.- 9. Core Concepts in Real Analysis.- A Summary of Strategies.- References.- List of Symbols. Index.