I am fairly new to writing proofs and am having difficulty understanding when to use a proof by contrapositive. A truth table clearly shows that $\neg Q\implies\neg P$ implies that $P\implies Q$. Given true propositions to prove, this is not much of a problem. But something like this is definitely not true:
Proposition: If $k$ is a positive integer, then $k \ge 5$.
Proof: Suppose $k$ is not a positive integer. Then $k$ is a negative integer. Since $k<0$, and $0<5$, it follows that $k<5$. Thus, $k$ is not greater than or equal to $5$.
Though the validity of this statement can easily be determined, I am not sure if I would be able to tell so quickly with a highly complex problem. Is there some sort of checklist to verify that a proof by contrapositive can be used?
Best Answer
Two immediate things:
The Proposition is not true ... and so it has no proof. But you could give a disproof: just point out that something like $k=2$ is a positive integer, but not greater or equal to $5$
Second, your attempt to prove the statement was not a proof by contrapositive. You showed $\neg P \to \neg Q$, rather than $\neg Q \to \neg P$