Posted by Anonymous on Sunday, September 28, 2008 at 10:01pm.
<<I am not converting them to "and" statements.>>
It seems to me that you are. The first clause of each statement has p and q statements separated by a symbol you say represents "or". The second clause of the sentence contains an "and" between separate conclusions about p and q.
I have reviewed the meaning of disjunction, and still believe your answers to both are wrong.
Put p = "Peter is a boy" and q = "Queenie is a girl". Then ~(p v ~q) means "It is not true that (either Peter is a boy or Queenie is a boy)". Doesn't that mean that both Peter is a girl and Queenie is a girl? If so, then p would be false and q would be true.
Similarly, ~(~p v q) means "It is not true that (either Peter is a girl or Queenie is a girl)". That would presumably imply that both Peter is a boy and Queenie is a boy. If so, then p would be true and q would be false.
In both instances you're effectively saying "If it isn't the case that at least one of A and B is true, then both of them must be false". (In the first instance, A=p and B=~q, whereas in the second instance, A=~p and B=q.)
I'm not absolutely certain my reasoning is correct here, but I think it is.
My reasoning is based on a truth table, does that change anything?
It ought not to: a truth table should deliver the same answers. I've just had a go at doing the first one that way, and got the following (these columns probably won't line up, but you should be able to work out which heading relates to which column):
P Q ~P ~Q (Pv(~Q)) ~(Pv(~Q)) (~P)^Q
T T F F T F F
T F F T T F F
F T T F F T T
F F T T T F F
This gives me the answer I got earlier (i.e. the last two columns are the same).
This might be easier to read - just ignore the underscores:
P_Q_~P_~Q_(Pv(~Q))__~(Pv(~Q)) __(~P)^(Q)
T_T__F__F______T__________F__________F
T_F__F__T______T__________F__________F
F_T__T__F______F__________T__________T
F_F__T__T______T__________F__________F
Ok. Thanks very much for all your help.