Suppose that the domain of the propositional function p(x) consists of −5, −3, −1, 1, 3, and 5. express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions. a) ∃xp (x) b) ∀xp (x) c) ∀x((x ≠ 1) → p (x)) d) ∃x((x ≥ 0) ∧ p (x)) e) ∃x(¬p (x)) ∧ ∀x((x < 0) → p (x))

a) ∃xP (x)

P(-5) v P(-3) v P(-1) v P(1) v P(3) v P(5)

(at least one of them is true)

b) ∀xP (x)

P(-5) ^ P(-3) ^ P(-1) ^ P(1) ^ P(3) ^ P(5)

(all of them are true)

c) ∀x((x ≠ 1) → P (x))

P(-5) ^ P(-3) ^ P(-1) ^ P(3) ^ P(5)

d) ∃x((x ≥ 0) ∧ P (x))

P(1) v P(3) v P(5)

e) ∃x(¬P (x)) ∧ ∀x((x < 0) → P (x))

[¬P(-5) v ¬P(-3) v ¬P(-1) v ¬P(1) v ¬P(3) v ¬P(5)] ^ [P(-5) ^ P(-3) ^ P(-1)]

[¬P(1) v ¬P(3) v ¬P(5)] ^ [P(-5) ^ P(-3) ^ P(-1)]

i believe it's a.

i'm not sure on this one, most likely correct.