I always thought that both “proof by reductio ad absurdum” and “proof by contradiction” mean the same, but now my professor asked this question on my homework and I don't know.
I believe that in both cases you assume the negation of the conclusion and develop a contradiction through the premises. This will imply the conclusion. Today I have a meeting with the assistant professor so I can clarify this, but I really would like to know what you guys think, or if possible it would be great if you point me into some good references.
I just came from my extra help and the assistant professor explains the difference this way:
Reductio ad absurdum: $$ \vDash [\neg p\to(q\wedge\neg q)]\to p$$
Proof by contradiction:
$$ \vDash [\neg (p\to q) \to (r\wedge \neg r)]\to (p\to q)$$
And the examples of application were these:
Using proof by contradiction: $\sqrt2$ is irrational.( First suppose it is rational and derive a contradiction).
Using proof by reductio ad absurdum: If $f$ is differentiable on $(a,b)$ then $f$ is continuous on $(a,b)$. ( First we suppose that $f$ is differentiable on $(a,b)$ but not continuous on $(a,b)$ and derive a contradiction).