Question

Let P(x), Q(x), R(x), and S(x) denote the following predicates with domain Z: P(x): x^2 + 2x - 15 = 0, Q(x): x is odd, R(x): x > 0, S(x): x^2 - 2 = 0. (a) For each predicate, determine its truth set. (b) For each of the following statements, determine whether it is true or false. If it is true, give reasons. If it is false, give a counterexample, i.e. determine a number x ∈ Z such that the statement does not hold for x. ∀x ∈ Z, P(x) → Q(x) (1) ∀x ∈ Z, (P(x) ∨ Q(x)) → R(x) (2) ∃x ∈ Z such that P(x) → (Q(x) ∧ R(x)) (3) ∃x ∈ Z such that ¬R(x) → P(x) (4) ∀x ∈ Z, S(x) → (Q(x) ∧ S(x)) (5) (c) Write down the negation of each statement in part (b).

          Let P(x), Q(x), R(x), and S(x) denote the following predicates with domain Z:
P(x): x^2 + 2x - 15 = 0,
Q(x): x is odd,
R(x): x > 0,
S(x): x^2 - 2 = 0.

(a) For each predicate, determine its truth set.
(b) For each of the following statements, determine whether it is true or false. If it is true, give reasons. If it is false, give a counterexample, i.e. determine a number x ∈ Z such that the statement does not hold for x.

∀x ∈ Z, P(x) → Q(x) (1)
∀x ∈ Z, (P(x) ∨ Q(x)) → R(x) (2)
∃x ∈ Z such that P(x) → (Q(x) ∧ R(x)) (3)
∃x ∈ Z such that ¬R(x) → P(x) (4)
∀x ∈ Z, S(x) → (Q(x) ∧ S(x)) (5)

(c) Write down the negation of each statement in part (b).

Added by Dana D.

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