Using proof by contradiction and a counter example. I have a vague question about proof by contradiction. The main goal of proofs by contradiction is to show that if some statement was true, then we would end up by a logical contradiction. My question would: in order to achieve the logical contradiction, would it be enough to just show a counter-example? That is we suppose the false statement, and then just give an example where the statement gives an illogical/absurd result. Would that be correct?

No that's not enought (if i understood what you mean). Think about it this way:

You want to prove A => B and you want to do it by contradiction. That means you suppose B is false even with the hypothesis A. Giving a counterexample to this means giving an example of A => B, and this doesn't prove A=>B in general.

For example: I try to prove that every odd number is prime (which is false); By contradiction suppose n is odd but not prime. I can find a counterexample to this (3 is odd and prime) but this doesn't give me the thesis (in fact 9 is odd but not prime)