Hi

I have to prove a simple proposition. But I am getting some problems

$\displaystyle (q \Rightarrow r) \vee (r \Rightarrow q) $

Now I can do this using truth tables. But I wanted to do it little differently. We can express disjunction as an implication.

$\displaystyle \neg (q \Rightarrow r) \Rightarrow ( r \Rightarrow q) $

since this is an implication, we can assume $\displaystyle \neg (q \Rightarrow r) $, so the goal is to prove $\displaystyle ( r \Rightarrow q) $. But this itself is an implication, so we can further assume , $\displaystyle r $ and so the goal is $\displaystyle q $ . So our givens are

$\displaystyle \neg (q \Rightarrow r) \mbox{ and } r $

and the goal is $\displaystyle q $. Now the givens give us

$\displaystyle \neg (\neg q \vee r ) \mbox{ and } r $

$\displaystyle \therefore (q \wedge \neg r) \mbox{ and } r $

So we get our goal, $\displaystyle q $. But I am also getting $\displaystyle \neg r $ and $\displaystyle r $. So I am little confused here.......