1. Proving a disjunction statement

Hi

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

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

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

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

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

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

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

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

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

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

2. Re: Proving a disjunction statement

Originally Posted by issacnewton
Hi
I have to prove a simple proposition. But I am getting some problems
$(q \Rightarrow r) \vee (r \Rightarrow q)$
Now I can do this using truth tables. But I wanted to do it little differently. We can express disjunction as an implication.
I am not sure that this is what you mean. But
\begin{align*} \left( {q \Rightarrow r} \right) &\vee \left( {r \Rightarrow q} \right) \\ \left( {\neg q \vee r} \right) &\vee \left( {\neg r \vee q} \right) \\ \left( {\neg q \vee q} \right) &\vee \left( {\neg r \vee r} \right)\text{ rearranged} \\T &\vee T\\ &T\end{align*}

3. Re: Proving a disjunction statement

Thanks Plato. Yes thats one way to do it. But I was trying to use the way like people prove math theorems which are given as an implication. So we assume antecedent and try to prove
consequent......... So there must be some error in my proof........