Intro Math Books for Learning Proofs | Jonathan

[NotGauss]

Hello all!
I am looking at books related to proofs, I have been looking around and it appears that either, "How it Prove It" or "Book of Proof" seem to be the top recommendations. Has anybody had any experience with these text and/or could provide some insight. Currently I am working through Apostol Vol. 1, will be starting Linear Spaces (Chapter 15) within a week or so and was thinking about doing a proof based book prior to going on to Apostol Vol. 2. Any input would be very appreciated.

Thank you for your time and help,
Jonathan

 

[dkotschessaa]

I enjoyed How to Prove it a lot. I don't remember the exact contents, but the second half (or so) of the book is pretty non-essential unless you want to go deeper into set theory.

 

[Logical Dog]

I have both in harcopy..they are excellent, I've finished 2/3 of book of proof. and about 20 pages of how to prove it :P

 

[MrRobotoToo]

Daniel Solow's https://www.amazon.com/dp/1118164024/?tag=pfamazon01-20 was a favorite of mine when I was a youngster.

 

[micromass]

I enjoyed How to Prove it a lot. I don't remember the exact contents, but the second half (or so) of the book is pretty non-essential unless you want to go deeper into set theory.

What is non-essential? Countable? Schroder-Bernstein? Induction? Injective?

 

[NotGauss]

@micromass, so that was a bit over my head

 

[dkotschessaa]

What is non-essential? Countable? Schroder-Bernstein? Induction? Injective?

Essential for learning how to write proofs? No (except induction, if it is not already known).

-Dave K

 

[micromass]

Essential for learning how to write proofs? No (except induction, if it is not already known).

-Dave K

No book is essential to write proofs. I could teach a person to write proofs in one day (and I have done that before with success). A book of proof tries to give essential prerequisite knowledge to be able to read any math book. Countable, injective, functions are all essential such knowledge. It's not only essential if you want to go into set theory, you'll use it virtually everywhere.

 

[dkotschessaa]

@micromass, so that was a bit over my head

Topics and definitions that you will certainly run into in your higher level/more abstract math classes. It is up to you if you want to learn them ahead of time. I never got that far in that particular book.

 

[NotGauss]

Should I go with a proof book prior to Apostol Vol II, currently or finish my Apostol sequence first. The proofs I have encountered thus far I can muscle through with internet searches but I feel as though I don't completely grasp them.

 

[micromass]

Should I go with a proof book prior to Apostol Vol II, currently or finish my Apostol sequence first. The proofs I have encountered thus far I can muscle through with internet searches but I feel as though I don't completely grasp them.

Doesn't really matter. A proof book won't help you much with Apostol, although what's inside it is very important for the rest of mathematics. So whether you do a proof book or Apostol II first is up to you. You could even work through them at the same time.

More imprtantly though, don't you have anybody to ask questions too, or somebody who can test your knowledge of the material? That is pretty essential. Way more essential than a proof book could ever be.

 

[Cruz Martinez]

Should I go with a proof book prior to Apostol Vol II, currently or finish my Apostol sequence first. The proofs I have encountered thus far I can muscle through with internet searches but I feel as though I don't completely grasp them.

I don't think it is essential to read a proof book in order to start doing math. If you're strugging with Apostol I doubt a proof book will help much, better to read another analysis book alongside Apostol or a naive set theory book such as the one by Halmos.

 

[NotGauss]

OK, thank you very much for the input!

 

[dkotschessaa]

No book is essential to write proofs. I could teach a person to write proofs in one day (and I have done that before with success).

That's great, and I'm glad it worked for those people. I personally found enormous benefit in learning logic, quantifiers, proof methods, etc. outside the context of a math course. A lot of people were very quick to tell me what a waste of time this was and I am glad to have ignored them.

A book of proof tries to give essential prerequisite knowledge to be able to read any math book. Countable, injective, functions are all essential such knowledge. It's not only essential if you want to go into set theory, you'll use it virtually everywhere.

So important are these topics that they will be covered ad nauseum in every upper level undergraduate math course. Learning them ahead of time is a great idea if one has the resources and time to do so. My point was that I didn't learn them from Velleman.

-Dave K