# How to Prove It: A Structured Approach, 2nd Edition

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Velleman wrote "How to Prove It": https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/...

Polya wrote "How to Solve It": https://www.amazon.com/How-Solve-Mathematical-Princeton-Scie...

awesomeopening that made me wish the math I was taught in school was done like this back when I went. Makes newer stuff make a lot more sense, too. I included a link to Dover that has a Google Preview button on it where you can read full, first chapter for free to see if it's what you like. Other two are more about exploring and proving things which may or may not interest you. I added them in case anyone is reading your question to learn that stuff.Concepts of Modern Mathematics by Stewart

https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

Dover Version with Google Preview Button

https://www.amazon.com/Introduction-Mathematical-Reasoning-N...

How to Prove It by Velleman

https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/...

[1] http://www.amazon.com/dp/0521675995/

[2] http://www.people.vcu.edu/~rhammack/BookOfProof/index.html

[3] http://www.amazon.com/dp/0982406207/

How to Solve Itwas recommended below. I also likeHow to Prove itby Daniel Velleman ().>Never done any proofs, but I will ask my math teacher if he can recommend any books or resources that will help me learn.

https://toptalkedbooks.com/amzn/0521675995

If you are interested in getting started with "real" math, i.e. moving from solving computational problems to writing proofs, I highly recommend picking up a copy of How to Prove It . The author assumes basically no knowledge of higher level math and teaches various techniques in reading and writing proofs, learning quite a bit of basic set theory along to way to practice your proof writing.

EDIT: wanted to also mention that set theory is the "core" of modern math, or rather its foundation.

I've always enjoyed all types of math but all throughout (engineering) undergrad and grad school all I ever got to do was computational-based math, i.e. solving problems. This was enjoyable but it wasn't until I learned how to read and write proofs (by self-studying How to Prove It ) that I really fell in love with it. Proofs are much more interesting because each one is like a logic puzzle, which I have always greatly enjoyed. I also love the duality of intuition and rigorous reasoning, both of which are often necessary to create a solid proof. Right now I'm going back and self-studying Control Theory (need it for my EE PhD candidacy but never took it because I was a CEG undergrad) and working those problems is just so mechanical and uninteresting relative to the real analysis I study for fun.

EDIT: I also love how math is like a giant logical structure resting on a small number of axioms and you can study various parts of it at various levels. I liken it to how a computer works, which levels with each higher level resting on those below it. There's the transistor level (loosely analogous to the axioms), the logic gate level, (loosely analogous set theory), and finally the high level programming language level (loosely analogous to pretty much everything else in math like analysis or algebra).