Proof methods and strategy

Question

1. When i want to solve some mathematical equation or problem, i need some proof methods and strategy. I know that there are some proof methods and strategy such as contradiction, mathematical induction , pigeonhole principle, inclusion-exclusion principle so on.

   I do not know that whether it is enough. I want to know all proof methods and strategy of mathematics such as contradiction,mathematical induction etc.

   Can anyone tell me what is the best book or website where i can learn all proof methods and strategy of mathematics ?

Answer 1

Just a comment: you can't expect anyone to ever know all proof methods and strategies of mathematics, as the art of mathematics consists to a great extent of understanding a problem well enough to come up with a working method.

However, what I think could be beneficial for you, is to look up a book in the genre of general introductions to higher mathematics, which may illuminate certain things. If you search online you might find some! Example: http://people.whitman.edu/~gordon/higher_math.pdf

Other things to consider could be to read proper textbooks on different suitable subjects. I for one think that the book on Discrete mathematics by Biggs has an excellent pedagocial introduction and structure which could be helpful to you. Another book that left a positive impression on me is A book of abstract algebra by Charles Pinter, which might be the best math book I've read in terms of readability.

Answer 2

You might want to work through How to Prove It: A Structured Approach, by $\underline{\mathrm{Daniel\; J. \,Velleman}}.$

I have a copy, in paperback, 2nd edition (2006). I can't tell you how many folks to whom I've recommended this book. (BTW I am not the author of the text!.) Compared to a lot of textbooks, this is a "steal" (relatively cheap and very high quality.)

I've found the pdf of of the complete book published on a faculty web-page for students to have access: How to prove it, Velleman. Look at the table of contents to see what you can learn with the text.

Note: Learning proof strategies requires far more than knowledge about proof strategies. You'll need to complement your studies "about" proofs with practice "doing proofs," of all kinds, in different contexts.