I'm trying to use De Morgan's Law to demonstrate an equivalence but I'm running into a problem when I try to apply it
De Morgan's Law states that ~(P V Q) == ~P ^ ~Q
However when I try to apply this to the following proposition:
~(~P V Q) == ~~P ^ ~Q == P ^ ~Q
I end up with truth tables that aren't equivalent
I'm sure I'm missing something obvious here, but I've been knocking my head against a wall trying to figure out what it is
Any help would be appreciated!
(Here are the truth tables I'm getting)
| P | Q | ~ P V Q | P ^ ~Q |
|---|---|---|---|
| T | T | T | F |
| T | F | F | T |
| F | T | T | F |
| F | F | T | F |
There is no problem actually. The fourth column is equivalent to the negation of the third column. The problem that you have, is that you forgot that you want $ \lnot (\lnot P \lor Q)$ and not $\lnot P \lor Q$.