Foundations of Higher Mathematics

by Peter Fletcher, C. Wayne Patty

This text introduces students to basic techniques of writing proofs and acquaints them with some fundamental ideas. The authors assume that students using this text have already taken courses in which they developed the skill of using results and arguments that others have conceived. This text picks up where the others left off -- it develops the students' ability to think mathematically and to distinguish mathematical thinking from wishful thinking.

Genres: Mathematics, Textbooks

336 pages, Hardcover

First published September 19, 1987

Reviews

[Sherri · Rating 4 out of 5]

P. Fletcher and C. W. Patty's book covers a wide array of concepts in higher math. Should definitely read it as an undergraduate math student, before taking a proofs class and other upper division math classes.

My advice is to focus on the first 5 chapters of the book. You will learn a lot more if you accompany your reading with other online texts (I used the free pdf : R. Hammack's Book of Proof) and videos (I used youtube : especially on the set theory section). This book really helped me develop an abstract thinking about math, through explaining why some theorems make sense.

The first 5 chapters give a solid introduction to logic, set theory, surjective/injective functions, and formal methods of proof (direct, contrapositive, contradiction). Important theorems in number theory are included- such as the fundamental theorem of algebra, division algorithm, chinese remainder theorem, and properties of congruence modulo n. The book also uses ¬P, rather than ~P, to denote the negation of P.

From Chapter 7 onward, the book is a bit rushed. It is understandable because the entire book is only 200 or so pages long. Chapters 8,9 introduce you to groups, subgroups, permutation groups, and abelian groups in abstract algebra, as well as the sequence and series convergence in real analysis. If you are self studying, you see that the abstract algebra chapter does not explain enough, and should to be supplemented with other reading material (I recommend Pinter's Abstract Algebra).

The exercises at the end of each section include computation and proof of lemmas and theorems. However, the solutions at the end of the book are not complete and only occasionally helpful.

Overall, it was a great introduction to higher math and logical thinking. Make sure to accompany it with other texts and online videos when you get confused!!

[Maya]

Good

[Salah Abuzaina · Rating 3 out of 5]

This book is a good base for advanced mathematical thinking. It explains the basics of mathematical proofs, and traces some main principles of mathematics,such as: set theory,mathematical induction,relations,functions,countable & uncountable sets and cardinal numbers.

[Brent · Rating 2 out of 5]

Annoying notation throughout the book. Inconsistencies in presentation. Annoying pseudo-mathematical terminology is used seemingly in an attempt to be humorous. Very light treatment of propositions and logic.

[Kris · Rating 4 out of 5]

This is a good primer on Mathematical thinking/proofs.