Rule of inference for proof by contradiction. In the book "Discrete Mathematical Structures" - Kolman, author has stated that proof by contradiction is based on the tautology ((p⇒q)∧(~q))⇒(~p).And that this argument form is often applied to the case where q is an absurdity.
But this tautology is modus tollens. 
In another text-book rule of inference for proof by contradiction is :
          ~p⇒c, where c is contradiction.
          ∴p

Please help me understand how rule of inference for proof by contradiction is modus tollens or based on above tautology. And what is the relation between two rules of inference?
 A: "Proof by contradiction" is, I take it, another label for the Reductio rule which can helpfully be displayed as
$$\quad\quad | \quad A$$
$$\quad\quad | \quad \vdots$$
$$\quad\quad | \quad C$$
$$\quad\quad | \quad \vdots$$
$$\quad\quad | \quad \neg C$$
$$\neg A$$
or (in another formulation)
$$\quad\quad | \quad A$$
$$\quad\quad | \quad \vdots$$
$$\quad\quad | \quad \bot$$
$$\neg A$$
When $A$ is a temporary assumption, which (via some subproof) leads to an explicit contradition or an absurdity $\bot$, we are allowed to discharge that temporary assumption and conclude (from whatever other premisses are in play) that it must be false, $\neg A$.
This is a valid rule of inference in systems of logic which lack a conditional (and even where a conditional can't be defined). So it is unhelpful -- to say the least -- to say that is "based on" a tautology involving a conditional (or on modus tolens).
If you do have reductio and modus ponens that modus tollens will be a derived rule:
$$A \to C$$
$$\neg C$$
$$\quad\quad | \quad A$$
$$\quad\quad | \quad C$$
$$\quad\quad | \quad \bot$$
$$\neg A$$
And conversely, if you have a conditional proof rule for introducing conditionals, modus tollens, and the assumption $\neg\bot$ then you could get reductio as a derived rule.

But, to repeat, it would be wrong to say that the result that (with a bit of help) you can get reductio from modus tollens is what "really" underlies reductio. Reductio is a warranted inferential rule because of the meaning of negation, not (even in part) because of the meaning of the conditional.

That, as they say, is the take-home message!
A: The thing you're missing is the law of non-contradiction:
$$ \neg (P \wedge \neg P) $$
i.e. $\neg c$ when $c$ is a contradiction.
To perform a proof by contradiction -- proving $\neg p$ via a proof of $p \implies c$ -- via proof by contrapositive, let $q$ be $c$.
The form of proof by contradiction you quote follows from the form I mention above by the equivalence $\neg \neg p \equiv p$. (so substitute $\neg p$ into your proof by contrapositive)
