Trying to find a contradiction for a certain proof (basic) I'm doing proof by contradiction for the first time and I understand the concept now, but I can't find a contradiction in my negation.
I'm trying to conclude $\neg n$
$h\wedge\neg r$
$(h\wedge n)\to r$

$~n$
My negations are:
$\neg h\vee r$
$h\wedge n\wedge\neg r$
 A: The negations of the two premises are not useful. You need to negate the sentence you are trying to prove, but the premises you use as they are, like this:
Assume $n$. From $h\wedge\neg r$ follows $h$. Infer $h\wedge n$. Use modus ponens to infer $r$. From $h\wedge\neg r$ follows also $\neg r$. Contradiction. Conclude $\neg n$.
A: *

*$h \land \neg r \qquad \qquad$ Premise

*$ (h \land n) \rightarrow r \qquad$ Premise

*$\qquad n \qquad \qquad \qquad$ Assumption (for the proof by contradiction)

*$\qquad h \qquad \qquad \qquad$ Simplification 1

*$\qquad h \land n\qquad \qquad$. Conjunction 3,4

*$\qquad r \qquad \qquad \qquad$ Modus Ponens 2,5

*$\qquad \neg r \qquad \qquad$ Simplification 1

*$\qquad \bot \qquad \qquad$ Contradiction 6,7

*$\neg n \qquad \qquad$ Proof by Contradiction 3-8
A: $$h\land \neg r$$  $$(h\land n) \rightarrow r$$

$$(h\land \neg r)\rightarrow\neg r$$
$$(h\land \neg r)\rightarrow h$$
$$\neg r \rightarrow \neg(h\land n)$$
$$(\neg(h\land n)\land h)\rightarrow\neg n$$
A: Remember, proof by contradiction works by assuming the negation of your conclusion, and inferring a contradiction - it has nothing to do with negating your premises.
Think about the problem in terms of the intuitive logic of the situation. Your premises say the following: first, "$h$ is true, and $r$ is not"; second, "if $h$ and $n$ are both true, then $r$ is true". Intuitively, if $n$ were true (the negation of your conclusion), we would have that $h$ and $n$ are both true (using the first half of the first premise). So $r$ is true (using the second premise). But $r$ is false (using the second half of the first premise), which means we now have $r \wedge \neg r$, a contradiction.
Now, the challenge is just taking those steps and converting them into formal logic.
