Let PQ mean "P and Q are both false." PQ is true if P and Q are both false, and false otherwise. Define the following logical connectives just using P, Q, , and parentheses.
1. ~P (not P)
2. P&Q (P and Q)
3. PVQ (P or Q)
4. P>Q (P implies Q)
5. P<>Q (P if and only if Q)
1. is quite simple. PP is false if P is true and true if P is false so ~P=PP. This simple negation will come in handy below.
Let the ordered quadruplet be the values of
P true Q true
P true Q false
P false Q true
P false Q false
So according to the problem we have
PQ=(f,f,f,t)
and we are seeking (I think. I've never really learned logic.)
2. P&Q = (t,f,f,f)
3. PvQ = (t,t,t,f)
4. P>Q = (t,f,t,t)
5. P<>Q = (t,f,f,t)
2. We want it to be true only if both P and Q are false. So we negate both and combine: (PP)(QQ)
3. Is the opposite of PQ so we just want to negate it: (PQ)(PQ)
4. If you combine the opposite of one with the other it will shift where the t is: (PP)Q = (f,t,f,f) so we just need to negate this:
((PP)Q)  ((PP)Q)
5. I had enough tables at this point to see that
(PQ)((PP)(QQ)) = (f,t,t,f) so we just need to negate this:
((PQ)((PP)(QQ)))  ((PQ)((PP)(QQ)))
I wouldn't be surprised if there were a simpler way to do this, but it seems to work for me.

Posted by Jer
on 20120124 13:58:26 