My question tries to address the intuition or situations when using the contrapositive to prove a mathematical statement is an adequate attempt.
Whenever we have a mathematical statement of the form A⟹B, we can always try to prove the contrapositive instead i.e. ¬B⟹¬A. However, what I find interesting to think about is, when should this approach look promising? When is it a good idea when trying to prove something to use the contrapositive? What is the intuition that A⟹B might be harder to do directly than if one tried to do the contrapositive?
I am looking more for a set of guidelines or intuitions or heuristics that might suggest that trying to use the contrapositive to prove the mathematical statement might be a good idea.
Some problems have structures that make it more "obvious" to try induction or contradiction. For example, in cases where a recursion is involved or something is iterating, sometimes induction is natural way of trying the problem. Or when some mathematical object has property X, then assuming it doesn't have property X can seem promising because assuming the opposite might lead to a contradiction. So I was wondering if there was something analogous to proving stuff using the contrapositive.
I was wondering if the community had a good set of heuristics for when they taught it could be a good idea to use the contrapositive in a proof.
Also, this question might benefit from some simple, but very insightful examples that show why the negation might be easier to prove. Also, I know that this intuition can be gained from experience, so providing good or solid examples could be a great way to contribute. Just don't forget to say what type of intuition you are trying to teach with your examples!
Note that I am probably not expecting an actual full proof super general magical algorithm because an algorithm like that could probably be used for automating prooving, which might imply something big like P=NP. (Obviously a proof of P vs NP is always interesting, but I think that asking the community to prove the P vs NP is not a realistic thing to ask...)
