Looking for Introduction to Proofs

In summary, the individual is seeking recommendations for an introductory book on proofs in Algebra and Trig, as they are struggling with the proofs in their current book, Basic Mathematics by Lang. They already have How to Prove It by Velleman, but are open to other suggestions. Other recommended books include "Book of proof" by Richard Hammack and Dexter Chua's lecture notes from Cambridge University.
  • #1
hackedagainanda
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Hey, PF! I am currently studying Algebra and Trig, and was wondering what's a good book to use to get eased into the process of proving statements and theorems. I'm planning to use Basic Mathematics by Lang but the proofs in it have been a real hindrance to my progress. Even just simple proofs like the irrationality of the square root of 2 are a struggle right now.
I currently have How to Prove It by Velleman, but was wondering if anyone has any experience with it or other introductory books on proofs.

My knowledge of Proofs is non-existent. Thanks for your time.
 
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  • #2
Usually this is covered in Geometry, e.g. "Geometry for Enjoyment and Challenge", R. Rhoad et al.
 
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  • #3
I think Velleman's book is highly regarded, you should try it. Another suggestion would be "Book of proof" by Richard Hammack. It is freely available online here, and the author, being a teacher, had prepared tests for a class he taught which you can find here (with solutions).
 
  • #4
"How to Prove It" is good. It has a nice summary of proof techniques at the end.

An alternative could be Dexter Chua's lecture notes. They're short, but they might also be too advanced for now. After all, it's from Cambridge University. But it's worth trying to explore that territory.

https://dec41.user.srcf.net/notes/IA_M/numbers_and_sets_trim.pdf
 

1. What is an introduction to proofs?

An introduction to proofs is a course or subject that teaches students how to construct and write mathematical proofs. It is an essential skill for any scientist or mathematician as proofs are used to validate and verify mathematical concepts and theories.

2. Why is learning proofs important?

Learning proofs is important because it helps to develop critical thinking and logical reasoning skills. It also allows scientists and mathematicians to communicate and understand complex ideas and theories using a standardized language and structure.

3. What are some common methods used in proofs?

Some common methods used in proofs include direct proof, proof by contradiction, and proof by induction. Direct proof involves using known axioms and definitions to logically arrive at a conclusion. Proof by contradiction involves assuming the opposite of what is being proved and showing that it leads to a contradiction. Proof by induction is a method used to prove statements about a sequence of numbers or objects.

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

To improve your proof-writing skills, it is important to practice regularly and seek feedback from peers or instructors. It can also be helpful to read and analyze well-written proofs to understand the structure and logic behind them. Additionally, familiarizing yourself with common proof techniques and strategies can also improve your skills.

5. Is an introduction to proofs only for math majors?

No, an introduction to proofs is not just for math majors. While it is a fundamental skill for mathematicians, it is also useful for scientists in various fields such as physics, computer science, and engineering. It can also benefit anyone looking to improve their critical thinking and problem-solving abilities.

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