Answer to Question #308962 in Discrete Mathematics for Zain

Question

The notation: ∃! x P(x)

means “There exists a unique x such that P(x)”.

If the domain consists of all integers, what are the truth values of these statement?

1. ∃! x(x > 1)

2. ∃! x(x

2 = 1)

3. ∃! x(x + 3 = 2x)

4. [∃! xP(x)] → [∃xP(x)]

5. [∀xP(x)] → [∃! xP(x)]

6. [∃! x~P(x)] → [~∀xP(x)]

7. ∃! x(x = x + 1)

8. ~(∃! xP(x)) → ∀xP(x)

9. (∃xP(x) ∧ ∃xQ(x)) → ∃x (P(x) ∧ Q(x))

10. (∀xP(x) ∨ ∀xQ(x)) → ∀x (P(x) ∧ Q(x))