[nicf]

I'm a mathematician, currently working as a private tutor for adults who want to learn proof-based math. I had a quick glance through this book and it seems to me like a pretty nice version of this "intro to proofs" sort of book. This is a topic that's done well in a lot of different books, though, so if you really want to dig into this topic I'd maybe recommend looking at a couple different ones and finding the one that agrees with you the most.

Right now I have a student working on this material and we're using "How to Prove It: A Structured Approach" by Daniel Velleman, which so far I'm finding decent. Some others I've seen (but that I haven't looked at in as much detail) are "Proofs: A Long-Form Mathematics Textbook" by Jay Cummings and "Book of Proof" by Richard Hammack.

[nyrikki]

As someone who tutors adults, can you suggest a more digestible book for abstract algebra?

While I was motivated, I used one of the typical college books. For me Abstract Algebra is what opened a lot of doors for me... but I am simply using applied math.

That moving away from proofs being magical across sub-topics is what I would like to share with some co-workers who are unwilling to buy a textbook and answer key.

As I didn't even mind Spivik for calc, my radar is way off for making suggestions to most people.

[tzs]

Pinter, "A Book of Abstract Algebra", is very nice. It's rigorous but not too terse. It divides the material into many small chapters with many exercises. Chapters are mostly around 10+/-3 pages with about 40-60% of that being text and the rest exercises.

The exercises for each chapter are split into several sections each section covering a different aspect of the chapter's material. Sometimes there is a section of exercises applying the material to some interesting area.

For example, the chapter on groups of permutations has 6 pages of text, then 5 pages of exercises divided into 9 sections. Those sections are: computing elements in S6 (5 problems), examples of groups of permutations (4 problems), groups of permutations in R (4 problems), a cyclic group of permutations (4 problems), a subgroup of SR (4 problems), symmetries of geometric figures (4 problems), symmetries of polynomials (4 problems), properties of permutations of a set A (4 problems), and algebra of kinship structures which consists of 9 problems covering how anthropologists have applied groups of permutations to describe kinship systems in primitive societies.

There are answers in the back for a decent number of the exercises.

It's a Dover republication so is not too hard on the wallet. List price is $30 at Dover but its around $20 on Amazon.

The combination of short chapters and lots of exercises make it easier than most textbooks to fit into a busy adult schedule.

[nyrikki]

Thank you, ordered it.

[sn9]

Abstract Algebra: A Student Friendly Approach by Dos Reis and Dos Reis [0] is like The Little Schemer but if it was a first course in abstract algebra.

[0] https://www.amazon.com/gp/product/1539436071?psc=1

[nicf]

I assume you're talking about an algebra book for self-study? Gallian's "Contemporary Abstract Algebra" is a common suggestion for a more accessible algebra book, and people also sometimes suggest Fraleigh's "A First Course in Abstract Algebra", but I can really only speak to what it's like to work on this stuff with a teacher --- since my students are by definition not self-studying the things I'm working on with them, my suggestions might be of limited use!

In general, I think self-studying proof-based math can certainly be done if someone's motivated enough, but it's pretty hard and takes a lot of work, especially if you're still getting used to the skill of reading and writing proofs. It's very valuable to be able to have a person available to evaluate the proofs you're writing, and I've definitely seen a few people who came to me thinking they'd mastered proof-writing on their own and were kind of mistaken about that. (I've definitely also seen people who really did learn this skill pretty well without help! It varies a lot.)

[nyrikki]

Thanks for the suggestion. I personally used Dummit and Foote's book and found it useful, but like early Calculus with Spivak, it seems most people prefer clear and concise over slightly more comprehensive and rigorous while still being introductions.

With self study I prefer a bit more breadth to make up for the realities of needing to self study which often ends up with deep but not wide understanding of topics.

Having a brother who had a PHD in complex analysis to bother probably helped with self-learning. That is the only option when you are on-call for decades at a time as higher math courses are/were always in person.

But hopefully someone will figure out a business model to help people who need to grow and adapt.

Thanks again for the suggestions, I have ordered both books to add to my lending library.

[sn9]

The 3rd edition of Velleman's book has an online supplement that uses Lean to work through the book so you can get feedback about your proofs.

[__rito__]

I have read many parts of Cummings'es book, and I can vouch for its quality.

It's the ideal book for learning proofs if you are self-learning.