It is a great irony of natural deduction that some of the most seemingly obvious inferences are also some of the trickiest to prove! So far, I haven't been able to prove the following, and I'd greatly appreciate if anyone has some nice elegant proofs for them:
~(P & Q) ⊢ ~P ∨ ~Q
~(P→Q) ⊢ P & ~Q
P ↔ Q ⊢ (P & Q) ∨ (~P & ~Q)
~(P ↔ Q) ⊢ (P & ~Q) ∨ (~P & Q)
I have a feeling that if I could get the first of these, the rest would fall out nicely.
PS — Is there a way of formatting logic here? Over at Mathematics they have nice formatting for logic, but I couldn't get it to work here.