TITLE: Using proof by contradiction vs proof of the contrapositive
QUESTION [173 upvotes]: What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by contradiction, and the other proves the contrapositive, the proofs look almost exactly the same.
For example, say I want to prove: $P \implies Q$
When I want to prove by contradiction, I would say assume this is not true.
Assume $Q$ is not true, and $P$ is true. Blabla, but this implies $P$ is not true, which is a contradiction.
When I want to prove the contrapositive, I say. Assume $Q$ is not true. Blabla, this implies $P$ is not true.
The only difference in the proof is that I assume $P$ is true in the beginning, when I want to prove by contradiction. But this feels almost redundant, as in the end I always get that this is not true. The only other way that I could get a contradiction is by proving that $Q$ is true. But this would be the exact same things as a direct proof. 
Can somebody enlighten me a little bit here ? For example: Are there proofs that can be proven by contradiction but not proven by proving the contrapositve?

REPLY [4 votes]: One difference is that proof by contrapositive only applies to propositions of the form $A \to B$ ("if-then propositions"). However not every proposition is a "if-then proposition", for example, consider the proposition, exist x real for all p,q integer, x != p/q, there is no $\to$ inside that proposition, so it is not feasible to prove it by contrapositive.
Proof by contrapositive:
$(\neg B \to \neg A) \to (A \to B)$
Proof by contradiction:
$(\neg A \to (B \land\neg B)) \to A $