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Is this a good textbook to learn proofs?

  • #1

Main Question or Discussion Point

Hey guys,

I was wondering if this textbook is good. I've seen the reviews and it has high stars. I want to learn proofs so I self-study and make my way to finally learning real analysis. I'm a chemistry major with a concentration in Math and economics. My plan was to do a double major in math and chemistry, but It would be too much to handle. Unless people have done it in the past and tell me about their experiences.

https://www.amazon.com/dp/0321797094/?tag=pfamazon01-20

https://www.amazon.com/dp/032174747X/?tag=pfamazon01-20 I will get this book later. First I want to learn how to read proofs and maybe do proofs.
 

Answers and Replies

  • #2
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In my limited experience, people start with Set Theory to learn the proper way to write proofs.

https://www.amazon.com/dp/1614271313/?tag=pfamazon01-20

I’m not sure if the Halmos book has any proofs in it but the related wiki description shows that it does

https://en.m.wikipedia.org/wiki/Naive_Set_Theory_(book)

Another book to consider is Proofs from The Book inspired by Erdos. It contains the most elegant proofs around.

https://www.amazon.com/dp/3662442043/?tag=pfamazon01-20

Here are a couple others with good reviews:

https://www.amazon.com/dp/0989472108/?tag=pfamazon01-20

https://www.amazon.com/dp/0521675995/?tag=pfamazon01-20

And lastly this recently published Reverse Mathematics book:

https://www.amazon.com/dp/0691177171/?tag=pfamazon01-20

I’ve only really looked at the Proofs from the Book which is more of a survey of the best proofs. Definitely something to keep around but because of its variety of proofs won’t teach how to do them well.

@fresh_42 or @Mark44 may have better recommendations.
 
  • #3
Math_QED
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What is your background? Did you finish high school? Are you still attending it? What math topics did you cover there? How familiar are you with proofs and what goals do you want to achieve?
 
  • #4
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I never really understood the necessity of these kind of books: How to prove? What's their purpose? Whom do they address? Personally I very much doubt that anyone, who has read a book of this kind, has better skills to solve the standard problem: "(Knowledge → Claim) now find the way!". To me it is obvious, that the size of "Knowledge" is the crucial part here, not the "→". Learning mathematics steadily confronts you with all kind of proofs, and to learn it means to follow the argumentation. If you can't learn proofs by reading proofs over and over, I don't see how a book should help. They are usually structured by the kind of proofs, so what is a categorization good for? To prove something is a creative process like painting. To know the names and kind of all brushes doesn't make you a painter.
 
  • #5
jasonRF
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I worked through the book by Lay (2nd edition) and thought it was quite good. The first couple of chapters cover set theory, logic, functions, methods of proof, etc. But the book is primarily a basic introduction to real analysis. Early chapter help walk you through the proofs, and as you get further into the book there is less hand-holding. So if you want to learn basic real analysis and don't have much experience with proofs it is not a bad choice. For me it was much more compelling to gain experience doing proofs in a context where I was also learning mathematics I cared about than to spend time with a "proof book".
 
  • #6
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To me it is obvious, that the size of "Knowledge" is the crucial part here, not the "→".
I would say not only the size, but the structure of knowledge. Kinda reminds me of Linus' opinion about good and bad programmers:
Bad programmers worry about the code. Good programmers worry about data structures and their relationships.
 
  • #7
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I never really understood the necessity of these kind of books: How to prove? What's their purpose?
Books like those mentioned can show the various styles of proofs: direct proofs, contradictions, contrapositives, if-and-only-if proofs, quantifiers, induction, and so on, plus examples of each. Two of these books that I've had for a long while are
"The Nuts and Bolts of Proofs," by Antonella Cupillari
"How to Read and Do Proofs, 2nd Ed." by Daniel Solow
The first class that many students encounter, that requires them to do proofs, is linear algebra. Many of them are unprepared for this kind of thinking, so struggle with how to present a logical argument.
 
  • #8
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The first time I hit proofs was in Geometry. The proofs were built u from earlier theorems that we proved. It gave us an understanding of the ned to backup your reasoning with an axiom or earlier taproot. In a sense, it helped us learn programming before we learned programming.

The problem was we became fluent in solving geometric proofs using geometric axioms but when we hit algebra then we needed to apply a new set of rules to our proofs and over time it becomes more difficult to figure out how to prove something as we switch to topology or some other math. @fresh_42 is right in that different proof strategies are used for different types of maths and the books I presented illustrate that.
 
  • #9
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Me, too, had this cultural shock at the transition from school to university and it's true, that some people gave up on it. I simply cannot imagine that there is a smooth way to do it other than in school itself. I think a book wouldn't have helped me. There was no way around digging calculus step by step, acre by acre. Are there more than induction, deduction, construction and contradiction and mixtures of them?
 
  • #10
The highest math courses I took in high school was Algebra. Then went to community college start at intermediate algebra, college algebra/Trig and Precalculus. Failed Calculus 1 once then the following semester I passed with a B. Transfer to a four-year university and took Calculus 2 ended up passing it with a B-. Currently taking calculus 3 right now. I want to learn proofs because I saw my friend taking a course in analysis and I always wonder what the fourmls come from. Since, many professors don't go over proofs and just give us the formula without telling us where is coming from.

That is why my interest came along. I want so self-study proofs and work my way to complex analysis. My goal is to be ready for a potential grad math course. I'm guessing I will have to take the GRE math subject test.

Like I said, if there are people who have done chemistry and Math at the same time please let me know.
 
  • #13
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What? Print it out chapter by chapter then or learn how to annotate PDFs.

It looks like a great resource. I’ve saved it for future reference.
 
  • #14
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Thank you, but I can't do PDF. I'm more of a textbook guy. Thanks anyways
The Hammack Book of Proof is also available in softcover from Amazon.
 

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