I just need help verifying my answers cause I'm still not 100% what I'm doing at the moment!
Let P and Q be predicates on the set S, where S has two elements, say,$ S = {a, b} $. Then the statement $ ∀xP(x) $ can also be written in full detail as $ P(a) ∧ P(b) $. Rewrite each of the statements below in a similar fashion, using P, Q, and logical operators, but without using quantifiers.
(b) $ ∃xP(x) ∧ ∃xQ(x) $ where I put $ (P(x)∨ P(y)) ∧ (Q(x) ∨ Q(y)) $ . I just was not sure because other example would describe it as $ ∃ x,y P(x) $ making it 2 variable. I just followed the procedure I know, just wanted feedback!
You need to use $a, b$ to get $$∃xP(x) ∧ ∃xQ(x) \equiv (P(a)∨ P(b)) ∧ (Q(a) ∨ Q(b))$$ since $x \in \{a, b\}$, and can take on either value.