Answer
a) ∀y∀x¬P(x, y) b) ∃x∀y ¬P(x, y)
c) ∀y(¬Q(y) ∨ ∃x R(x, y))
d) ∀y(∀x¬R(x, y) ∧ ∃x¬S(x, y))
e) ∀y(∃x∀z ¬T(x, y, z) ∧ ∀x∃z ¬U(x, y, z))
Work Step by Step
We need to use the transformations shown in Table 2 of Section 1.4, replacing ¬∀ by ∃¬, and replacing ¬∃
by ∀¬. In other words, we push all the negation symbols inside the quantifiers, changing the sense of the
quantifiers as we do so, because of the equivalences in Table 2 of Section 1.4. In addition, we need to use De
Morgan’s laws (Section 1.3) to change the negation of a conjunction to the disjunction of the negations and to
change the negation of a disjunction to the conjunction of the negations. We also use the fact that ¬¬p ≡ p.
