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How to Prove It: A Structured Approach

Daniel J. Velleman · 3 HN comments
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Amazon Summary
Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.
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  • Ranked #14 this year (2022) · view

Hacker News Stories and Comments

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I have not read it, but have often seen this one recommended:

How to Prove It: A Structured Approach by Velleman. New edition came out in 2019. It appears to be aimed at your level, and pricewise isn't too bad.

I couldn't disagree more. I studied Analysis in the Uni, and even in that environment Rudin is pretty bad. For a total newcomer that book will leave you completely helpless. Also, solutions are a must have, without them you are almsot totally lost. In their absence, it is OK to ask on StackExchange or #math on EFnet.

First let's start with a few books to prep you for college-level maths:


* ; or

* ; or

* (I believe you can find solutions to the 2nd edition online)

For Single-Variable Analysis



* (contains solutions to exercises)

* (there are solutions online for the 2nd edition)

* (this book is a brilliant exercise-guided approach that helps you build up your knowledge step by step + solutions are provided).

Agree, Rudin is excellent as a second course on real analysis, but as a first one it is absolutely terrible.
+1, but nitpicking: There are situations where baby Rudin ("Principles of Mathematical Analysis") can be very good for a first course...but those are corner cases, that you'd want a really good, experienced instructor to sign off on.
I heard this from a senior colleague of mine that he's been working through the book "How to prove it: A Structured Approach"[1] that show how to prove things in mathematics, and has quite good exercises.

Perhaps a hands-on approach such as solving the exercises while working through this book will prove beneficial and is complimentary to watching, say, KA or 3B1B.


I think that book is better once you're already fairly comfortable with proofs. I tried to read it early on in my mathematics studies and couldn't understand it. Until you've done a number of proofs and start to see commonalities between them (e.g. proof by contradiction, induction, etc.) you don't have the mental scaffolding to learn more systematic approaches to proofs. I realize the book attempts to teach these techniques, but in my opinion it is hard to motivate these until someone has actually needed to use them to solve problems they're interested in.
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