Logical Equivalences with quantifiers
I am running into problems with the following:
∃x(P (x) ⇒ (Q(x) ⇒ R(x)))⇐⇒ (¬∀xP (x) ∨ ¬∀yQ(y) ∨ ∃zR(z))
Although I do not master logical equivalences, I have been able to solve some earlier. However, since this problem has quantifiers, I don't know how to approach it.
Any help is highly appreciated.
Re: Logical Equivalences with quantifiers
P(x) => (Q(x) => R(x)) is equivalent to ~P(x) \/ ~Q(x) \/ R(x).
Existential quantifier distributes over disjunction (because existential quantifier is basically a disjunction over all elements of the domain): .
Next, existential quantifier changes into universal quantifier and vice versa when it is moved through a negation: .
These equivalences are sufficient to derive the one you need.