Help on propositional calculus problem Prove that the statements “$(p \mathbin{\text{and}} \neg q) \mathbin{\text{implies}} q$” and “$(p \mathbin{\text{and}} \neg q) \mathbin{\text{implies}} \neg p$” are logically equivalent. What simpler statement is logically equivalent to both of them?
 A: The following are equivalent:


*

*$p\wedge q\Rightarrow \neg q$

*$\neg \left(p\wedge q\right)\vee\neg q$

*$\left(\neg p\vee\neg q\right)\vee\neg q$

*$\neg p\vee\neg q$


Starting at 1) you arrive at 4). If you start with $p\wedge q\Rightarrow \neg p$
then you will end up with the same result.
A: Maybe this would help.  Take the statement $(A\wedge B)\Rightarrow -B$.  Then to obtain your first statement let $A=p$ and $B=-q$.  Then use that $--q=q$.  For your second statement let $A=-q$ and $B=p$.  Here you need to use that $A\wedge B=B\wedge A$.
A: You might assume them equivalent, then only using equivalences try to find a common form.  I use Reverse Polish Notation.  Thus, instead of (p$\land$q) I write pqK.  Instead of (p$\lor$q) I write pqA.  Instead of $\lnot$p, I write pN.  Instead (p$\rightarrow$q) I write pqC.  I'll also write the logical equivalence of $\alpha$ and $\beta$ as
$\alpha$ $\beta$ E
So, let's suppose  


*

*pqNKqC  and pqNKpNC logically equivalent or "pqNKqC pqNKpNC E".  Since pqC pqNKN E we can thus obtain 2 from 1.

*pqNKqNKN pqNKpNNKN E.  Since ppN p E we can obtain 3.

*pqNKqNKN pqNKpKN E.  Since pqKN pNqNA E we can obtain 4.

*pqNKNqNNA pqNKNpNA E.  Since ppN p E we can obtain 5.

*pqNKNqA pqNKNpNA E.  Since pqKN pNqNA E we can obtain 6.

*pNqNNAqA pNqNNApNA E.  Since q qNN E we can obtain 7.

*pNqAqA pNqApN E.   Since pqArA pqrAA E we can obtain 8.

*pNqqAA pNqpNAA E.    Since pqA qpA E we can obtain 9.

*pNqqAA pNpNqAA E.    Since pqrAA pqArA E we can obtain 10.

*pNqqAA pNpNAqA E.   Since ppA p E we can obtain 11.

*pNqA pNqA E.
Consequently, since we only used equivalences, and since equivalences all come as reversible when used, we could have started with 11. and derived 1 using the equivalence indicated on the right in the reverse order.  11. also contains a statement form logically equivalent to both of the original two statement forms. 
