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Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition) (Featured Titles for Transition to Advanced Mathematics)

Gary Chartrand, Albert D. Polimeni, Ping Zhang · 4 HN comments
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Amazon Summary
Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.
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First read

By one of my early mathematics tutors in San Diego math circle

Then buy something like: Mathematical Proofs: A Transition to Advanced Mathematics (3rd Edition) (Featured Titles for Transition to Advanced Mathematics)

Also art of problem solving vols 1-2 are now classics for that age.
My answer to your question is math. Learn to read and write proofs. Any intro to proofs will do: those employed in discrete math, the ones in analysis, the diagram chasing ones, whatever...Working with math proofs will definitely straighten out your thinking and whip your mind into shape.

Some suggestions to get you started:

Book of Proof by Richard Hammack:

Discrete Math by Susanna Epp:

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al:

How to Think About Analysis by Lara Alcock:

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers:

Mathematics: A Discrete Introduction by Edward Scheinerman:

The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Rafi Grinberg:

Linear Algebra: Step by Step by Kuldeep Singh:

Abstract Algebra: A Student-Friendly Approach by the Dos Reis:

That's probably plenty for a start.

great, thank you!
A more graduate-level book but one I found pleasing:

A Logical Approach to Discrete Mathematics:

And a more pragmatic approach to the same material (with a lot of cross-over in terms of proof-style, etc):

Programming in the 1990s:

But one I particularly enjoyed early on was written for liberal-arts level students of maths (who might've been traumatized by maths in the past):

Introduction to Graph Theory:

It will actually get you into writing proofs in set theory within the first couple of chapters.

Oh gosh the equational logic rabbit hole.

To add to the fire:

I agree that getting things in the right order is important, but would argue that the order in which math is usually taken in the US is not the optimal one! I recently took calc-1,2,3 and linear algebra through my local community college, and then started working my way through a wonderful book on mathematical proofs:(, as preparation for working on higher level math. I would now argue that being able to understand and write proofs is a (the?) key mathematical skill to understanding what I would call 'real' (higher) math, and could be learnt by most students following high school algebra. My impression of the calculus series and linear algebra courses was an excessive focus on calculation, the math proof book was way more fun, surprising for a subject that is often thought to be too difficult for first year college students. For those who are intimidated by the idea of a book on proofs (like I used to be), an example from the third chapter:

Theorem: Let x be an integer. Then x^2 is even if and only if x is even

It seems so simple, and I think would be accessible to anyone who had completed high school algebra but I found that even having done those calculus and linear algebra courses, I had now idea how to go about actually PROVING this! The book however, goes through the thought process step by step, and teaching the skills needed to be able to understand the real math books like Rudin.

You might enjoy one of the many books that exist for undergraduates to ease the transition into higher math classes, where there is a shift from the strong calculation-focus and rote-learning of most math teaching through calculus, to the more proof-centric and understanding-based approach that one finds in classes like abstract algebra, real and complex analysis, and other post-calculus math courses.

Here are some examples of the kinds of books I mean, and you can find others by following Amazon recommendations from those:

With math textbooks especially, it pays to look for a previous edition, as the current edition can be ridiculously expensive, and the previous edition might be only 20% of the price, with no significant differences between the two.

Also, don't get them for the Kindle, as Amazon doesn't seem capable of publishing a math book with lots of notation that doesn't also have tons of errors where symbols get incorrectly imported. I've bought at least 20 and yet have to see one that didn't have lots of incorrect symbols.

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