Please help me solve these questions.
Question 1 : Determine whether following propositions are equivalent. Justify your answers.
1. ∀x (P(x) → Q(x)) and ∀x (P(x)) → ∀x (Q(x)).
2. ∀x (P(x) ⇔ Q(x)) and ∀x (P(x)) ⇔ ∀x (Q(x)).
Question 2 : Show that ∃x (P(x) v Q(x)) and ∃x(P(x)) v ∃x(Q(x)) are logically equivalent.
Question 3: Establish these logical equivalences, where x does not occur as free variable in A. Assume that the domain is nonempty.
a) ( ∀x P(x)) ∨ A) ≡ ∀x (P(x) ∨ A)
b) (∃x (P(x)) v A) ≡ ∃x (P(x) v A)
Question 4: What is the truth value of this statement.
∃x! (P(x)) →¬ ∀x P(x)
Do provide explanation wherever possible
Thank You.
