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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))

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Answer:

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)]

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