Transition to Math Proofs: Tips & Books for Upper Level Courses

In summary, the conversation discusses the transition from computational math courses to proof-based courses and asks for book recommendations. Several books are suggested, including Zorich Analysis I, Serge Lang's Introduction to Linear Algebra, Krantz & Bellman, Lay, and Epp's Discrete Mathematics. The individual also mentions using logic and implications to understand proofs and recommends the book "The Art and Craft of Problem Solving" by Paul Zeitz. They also recommend Stephen R Lay's book as a good transition to more rigorous mathematics.
  • #1
IKonquer
47
0
Hi all, I have taken Calc III, Linear Algebra (Bretscher's book), and an ODE class, which have all been mostly computational. I plan on taking upper level math courses such as abstract algebra and analysis, and my understanding is that the latter are proof based rather than computational. Are there any good books out there that can help me make that transition to more abstract ideas and proofs?

Thanks in advance.
 
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  • #2
https://www.amazon.com/dp/007154948X/?tag=pfamazon01-20 is a lifesaver, as was the first chapter of
Zorich Analysis I as well as the first chapter of Serge Lang's
Introduction to Linear Algebra
. Another phenomenal book is https://www.amazon.com/dp/0394015592/?tag=pfamazon01-20
Basically the greatest discovery I've had this year was to realize how proofs follow from
logic, i.e. implications, a chain of implications, logical equivalences etc... I think the first few
chapters of the Krantz book will give you the idea then you should do some further
research into how to use these ideas, another book that uses these ideas very well is
https://www.amazon.com/dp/0131481010/?tag=pfamazon01-20. The most preferable thing, to me, would be to do
Krantz & Bellman along with the first chapter of Lang, then the first few chapters of Lay.
Finish it off by doing the first chapter of Zorich & you'll be where I am now, trying to get
better :tongue: Before all of this I was extremely insecure about proofs, struggling to
understand the "logic" behind any of it & struggling to find patterns but now it's an
enjoyable experience of turning authors "wordy" proofs into a chain of logical implications,
well - assuming they are not too advanced or just incomprehensible to me :shy:

Note: These are personal preferences from experience, honestly all you need is Krantz,
Bellman & Lay as each gives insight I have not found in a single other book after
mercilessly searching,. The Lang chapter is just so beautiful as so much of the chapter is
derivable from a single chain of logic:

1) Take two vectors A & B 2) Make B longer than A (see page 23) 3) Find a vector orthogonal to B. 4) (A - cB)•B = 0 5) Use Pythagorean Theorem 6) ||A||² = (||A - cB||)² + ||cB||²
7) Prove that ||cB|| = |c|||B|| 8) ||cB||² = (√(cB)-(cB))² = c²B-B = c²||B||² 9) ||cB||² = c²||B||² ⇒||cB|| = |c|||B|| 10) ||A||² = (||A - cB||)² + |c|²||B||²
11) Notice c²||B||² ≤ ||A||² 12) Derive c 13) (A - cB)•B = 0 14) A•B - cB•B = 0 15) A•B = cB•B 16) c = (A•B)/(B•B) 17) c²||B||² ≤ ||A||² → 18) [(A•B)/(B•B)]² ||B||² ≤ ||A||²
19) [(A•B)/||B||²]² ||B||² ≤ ||A||² 20) [(A•B)²/||B||²] ≤ ||A||² 21) (A•B)²≤ ||A||²||B||² 22) A•B ≤ ||A||||B|| 23) C•C ≤ ||C||||C|| 24) Derive the Triangle Inequality Yourself!

(Sig on another forum :redface:).

Another list of books worth researching, as regards getting used to proofs, are:
Gleason - Fundamentals of Abstract Analysis
Maddox - Transition to Abstract Mathematics
Morash - Bridge to Abstract Math
Epp - Discrete Mathematics
Grimaldi - Discrete Mathematics

These looked like the best choices to me, hope this helps somewhat!
 
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  • #3
to learn to write proofs I used a book called 'the art and craft of problem solving' by paul zeitz. It's pretty informal... but it's great. It isn't specifically about proof writing but the entire book will help you, with proofs and a whole lot more
 
  • #4
I personally think Stephen R Lay's book is a good transition to more rigorous mathematics.
First few chapters are really recommended for people need to understand more on proofs.
 
  • #5


I can understand your concern about transitioning from computational math to proof-based upper level courses. One suggestion I have is to start by familiarizing yourself with the basic principles and techniques of proof writing. This can include understanding logical reasoning, using mathematical notation effectively, and identifying common proof structures. There are many resources available online and in book form that can help with this, such as "How to Prove It" by Daniel Velleman or "How to Read and Do Proofs" by Daniel Solow.

Additionally, it may be helpful to start practicing with simple proofs in your current courses or on your own. This will help build your confidence and prepare you for the more abstract and rigorous proofs in upper level courses. You can also seek out a mentor or join a study group to discuss and work through proofs together.

In terms of specific books for upper level courses, it would be beneficial to consult with your professors or academic advisors who can recommend textbooks that align with the specific course material. Some popular books for abstract algebra include "Abstract Algebra" by David Dummit and Richard Foote and "A Book of Abstract Algebra" by Charles C. Pinter. For analysis, some recommended texts are "Principles of Mathematical Analysis" by Walter Rudin and "Real Analysis" by Royden and Fitzpatrick.

Overall, the key to successfully transitioning to proof-based courses is to practice and seek guidance from experienced mathematicians. With dedication and perseverance, I am confident that you will excel in these challenging but rewarding courses. Best of luck to you in your studies!
 

1. What are some tips for transitioning to math proofs in upper level courses?

Some tips for transitioning to math proofs in upper level courses include practicing regularly, understanding the basic principles and concepts, breaking down the proof into smaller steps, and seeking help from peers or professors when needed.

2. How can I improve my proof-writing skills?

To improve proof-writing skills, it is important to read and analyze existing proofs, break down complex proofs into smaller steps, and practice writing proofs regularly. It is also helpful to seek feedback and suggestions from peers or professors.

3. What are some recommended books for transitioning to math proofs?

Some recommended books for transitioning to math proofs include "How to Prove It: A Structured Approach" by Daniel J. Velleman, "Mathematical Proof and Structures" by Larry J. Gerstein, and "Introduction to Mathematical Structures and Proofs" by Larry J. Gerstein and Bruce H. Arnold.

4. What are some common mistakes to avoid when writing proofs?

Some common mistakes to avoid when writing proofs include not clearly stating the assumptions and definitions, making incorrect assumptions, using incorrect logic or reasoning, and not clearly explaining each step of the proof.

5. How can I better understand the logic and structure of proofs?

To better understand the logic and structure of proofs, it is helpful to study and practice various proof techniques, such as direct proof, proof by contradiction, and proof by induction. It is also important to understand the definitions and assumptions used in the proof and to break down the proof into smaller, more manageable steps.

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