If a problem asks me to give a counterexample to a conjecture, am I proving the conjecture true or false by giving a counterexample? I am leening towards proving it false, because if I were to prove it true, it would involve another technique (contradiction, direct, ... )
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A counterexample to a statement shows that it is false, while a proof shows that it is true.
If you can find a counterexample, then you are refuting the conjecture, therefore proving it wrong!! If you can prove the conjecture with logical arguments(rigurously), then the conjecture is not a conjecture any more, and it becomes a theorem, or proposition!