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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$\begingroup$ One little thing, existential statements can be proven true by example(e.g. I can prove "there exists a even prime" by giving you the example of 2, which is even and prime), but if we use the prefix "counter" it means the example will prove the negation of the statement, or equivalently disprove the statement. $\endgroup$– Sean EnglishNov 8, 2015 at 18:59
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2 Answers
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A counterexample to a statement shows that it is false, while a proof shows that it is true.
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$\begingroup$ Thanks, the reason I was beginning to become confused is because after many examples and thinking, intuition indicates that the conjecture is true(i.e a theorem). $\endgroup$ Oct 23, 2012 at 12:51
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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!
answered Oct 23, 2012 at 12:49