How can you effectively approach writing a proof?

[MiniTank]

I find proofs very difficult. What process would you go through before writing a proof? What do you do generally?

 

[HallsofIvy]

There isn't any "general theory" or "general procedure"- you have to THINK!

I would recommend the following- first write down the hypotheses (what you are "given") and the conclusion (what you are trying to prove). Now, write down the definitions of all technical words (Yes, I think it is a really good idea to WRITE them just to be sure you have the precise definition- often specific words of the definitions are used in proofs) and any formulas you can think of (or look up!) connecting any quantities given in the proof. NOW try to think of all the ways in which the hypotheses could be connected to the conclusion. Often it helps to look at a simplified version of the proof first- are there any special cases in which the proof is simpler? If so can you vary the proof so as to apply to the more general case?

 

[thepatientmental]

Before writing a proof, it is important to have a clear understanding of the concepts and definitions involved in the problem. This includes reviewing any relevant theorems or properties that may be applicable. It is also helpful to carefully read and analyze the given statement or question to determine what exactly needs to be proven.

Once I have a solid understanding of the problem, I like to start by brainstorming and outlining my approach to the proof. This may involve breaking down the statement into smaller, more manageable parts or considering different strategies for proving it. I also like to consider any counterexamples that may disprove the statement and think about how to address them in my proof.

Next, I begin writing the actual proof, making sure to clearly state each step and justify it using previously established definitions or theorems. It is important to be thorough and meticulous in the logical progression of the proof, as even a small mistake can lead to an incorrect solution.

Throughout the process, I also like to revisit my work and check for any errors or gaps in reasoning. It can be helpful to take breaks and come back to the proof with a fresh perspective, as this can often reveal errors that may have been overlooked.

Overall, my general approach to writing a proof involves careful preparation, strategic thinking, and thorough checking for accuracy. With practice and patience, proofs can become less daunting and more manageable.