Sets, Functions and Logic: Basic concepts of university mathematics

by Keith Devlin

The purpose of this book is to provide the student beginning undergraduate mathematics with a solid foundation in the basic logical concepts necessary for most of the subjects encountered in a university mathematics course. The main distinction between most school mathematics and university mathematics lies in the degree of rigour demanded at university level. In general, the new student has no experience of wholly rigorous definitions and proofs, with the result that, although competent to handle quite difficult problems in, say, the differential calculus, he/she is totally lost when presented with a rigorous definition oflimits and derivatives. In effect, this means that in the first few weeks at university the student needs to master what is virtually an entire new language {'the language of mathematics'} and to adopt an entirely new mode ofthinking. Needless to say, only the very ablest students come through this process without a great deal of difficulty.

Genres: Mathematics, Logic

Paperback

First published January 1, 1981

About the author

Dr. Keith Devlin is a co-founder and Executive Director of the university's H-STAR institute, a Consulting Professor in the Department of Mathematics, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 26 books and over 80 published research articles. Recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is "the Math Guy" on National Public Radio.

Reviews

[Richard Newton · Rating 4 out of 5]

I approached this book with a little trepidation as it is some time since I have done any real mathematics. However, I am doing a philosophy degree I wanted to improve my use of logic and understanding of set theory. It is not exactly a light read, but that would not be expected. It is a well structured, if brief, introduction to sets, functions and logic.

Take time to read it with care and go through the exercises and the book does what it sets out to achieve. The author has kept it short. It reflects his experience teaching mathematics as the contents are clear and sufficiently comprehensive. Choose this if you are a reasonably confident student who wants a friendly, but concise introduction to these topics - or if you are looking for a refresher to bring stale knowledge back to life. If you are really starting from zero knowledge you may find this a little too concise.

The book is targeted at those who have done calculus and are moving onto the next level of mathematics. It is a long time (over 20 years) since I did any calculus, and yet I found this book reasonably straightforward to follow and did not find myself having to recall much from those distant times!

It would have been better for me if the exercises had answers. The author leaves these out, reasoning this is not a book for individual study, but to be used as part of a course and as such you should discuss your answers with colleagues and tutors. However, this is not a fatal problem and I used the book successfully for independent study.

[Anartam · Rating 5 out of 5]

Sets, Functions, and Logic is a lovely short book written for students preparing to attend Pure Mathematics classes. It's supposed to be a transition between calculus and courses on Abstract Algebra, Analysis, Math Fundamentals.
The first chapter provides an overview of Contemporary Math, presenting some historical perspective.
The second chapter introduces math notation very gently. Some concepts from mathematical logic are presented in a friendly way, as well as a raw account of mathematical proofs. The topic on negation and quantifiers is particularly good.
The third chapter deals with the basics of Set Theory, and it is straightforward and objective. Devin adds a section on the history of set theory, which is enlightening.
The fourth chapter focuses on Functions. It also adds discussions on denumerability and uncountability. Since readers are students making a transition to Pure Mathematics, those topics are a bit harder to grasp. The reader can find a clearer presentation of those topics in Barbara Partee, Alice ter Meulen, and Robert E. Wall’s book Mathematical Methods in Linguistics.
Finally, the fifth and last chapter is about Relations. Devlin chose to present Functions before the chapter on Relation, which is an unusual choice. The section on Upper Bounds and completeness is the most challenging topic in the chapter.
Something to consider: this book isn't intended for self-study. It provides no answers to exercises. Exercises are used as a tool to sharpen readers' math skills and develop some concepts. Readers need to study in a group and consult the instructor for feedback.
Overall, this book is an excellent introductory book. However, the reader will probably need other introductory books to grasp the content fully. How to prove it, by Daniel Velleman, would be an excellent complementary source.

[Carol · Rating 2 out of 5]

2.5 stars
Not much new here. The book was intended for college students between calculus and analysis. I should have read it back in the 60s

[Mahin · Rating 3 out of 5]

A decent book. (∃ Better book) ∧ (∃ Worse book) than this

[Nicholas Piercy · Rating 5 out of 5]

Excellent primer for learning proof-based logic and abstract mathematics. Great for the student entering mathematical analysis.

The author does an impeccable job of demystifying the mathematical basis by incorporating real world explanations in-line with the text.

This book changed the way I think, and has had an immeasurable impact on my studies.

Highly recommend!