A Transition to Proof: An Introduction to Advanced Mathematics

by Neil R. Nicholson

A Transition to Proof: An Introduction to Advanced Mathematics describes writing proofs as a creative process. There is a lot that goes into creating a mathematical proof before writing it. Ample discussion of how to figure out the "nuts and bolts'" of the proof takes place: thought processes, scratch work and ways to attack problems. Readers will learn not just how to write mathematics but also how to do mathematics. They will then learn to communicate mathematics effectively.

The text emphasizes the creativity, intuition, and correct mathematical exposition as it prepares students for courses beyond the calculus sequence. The author urges readers to work to define their mathematical voices. This is done with style tips and strict "mathematical do’s and don’ts", which are presented in eye-catching "text-boxes" throughout the text. The end result enables readers to fully understand the fundamentals of proof.

Features:

The text is aimed at transition courses preparing students to take analysis

Promotes creativity, intuition, and accuracy in exposition

The language of proof is established in the first two chapters, which cover logic and set theory

Includes chapters on cardinality and introductory topology

Genres: Mathematics

462 pages, Kindle Edition

Published March 21, 2019

Reviews

[Tom Schulte · Rating 4 out of 5]

"Generally, I find works promising to guide the reader in constructing proofs and approaching advanced mathematics reside at either end of a spectrum. At one end, at least an advanced undergraduate is the target, at the other the material is too lacking in rigor for serious merit. Here, easily two thirds of the material on symbolic logic and writing proofs has value to the high school student planning to major in mathematics as well as correct any bad habits in the aforementioned advanced undergraduate..."

[Look for my entire review at MAA Reviews]

[Renjie · Rating 4 out of 5]

The book is very in details, explaining the ideas of constructing proofs in mathematics. The examples are mainly taken from set theory, and number theory. I like the clarity, but I would prefer it to be more concise.