What are the proper steps needed to solve a prove question?

Question

I was just wondering is there any particular structure of steps you take to solve a question about proof.

For eg, For each pair of real numbers x and y, if x + y is irrational, then x is irrational and y is irrational.

Since this question is relatively straightforward, many people are able to tell right away that it is a false statement and come up with a counter-example. However, what if there is a question where it's not so straightforward?

Must I manipulate the statement to find its negation, contrapositive etc and from there start to gauge if the statement is true or false? How would I know when to use negation and when to use a contrapositive? More precisely, what are the steps you take in trying to prove difficult statements? Because when I come across a question like this, I am just completely stunned and have no clue where to start.

Answer 1

The only really good questions are the ones for which there are no "proper steps" that lead to an answer.

You get better at "prove or disprove" questions by doing lots of them.

Start with your first instinct. If you think it's false, then carefully look at the hypotheses and the conclusion to try to construct a counterexample.

If you spend some time on that and don't find one, then perhaps the assertion is true. So change strategies and look for a proof. Is there a simpler case where you can find one? Are there consequences of the hypotheses that look promising? Can you rephrase the conclusion in terms that seem more intuitive or make more sense to you? Are there related theorems whose proofs you might mimic?

If you don't find a proof, maybe the result is false after all. By looking at the places where your proof got stuck you might find the counterexample. After a while failing to find one you go back to trying for a proof, using what your failure told you as a guide.

Real mathematics is this kind of back and forth, trying to find out what is actually true about these abstractions. The more you study, the more examples you know, the better your instincts get about what is true and what strategies help you know.