For each proposition below, decide whether it is true or false and give a brief explanation. Assume the universe (domain of variables) to be Z, the set of integers. (1) ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0)) (2) ∀x((x < 0) ∨ (x^2 ≥ x)) (3) ¬(∃xP(x)) ↔ (∀x¬P(x)) for all predicates P(x)

For each proposition below, let us decide whether it is true or false. Assume thedomain of variables to be $\Z$, the set of integers.

(1) $((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0))$

The conjunction $(x = 5) ∧ (y = 1)$ is true if and only if $x=5$ and $y=1.$ In this case the disjunction $(x > 10) ∨ (y > 0)=(5 > 10) ∨ (1 > 0)$ is true, and hence the implication $((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0))$ is also true. In other cases the conjunction $(x = 5) ∧ (y = 1)$ is false, and hence the implication $((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0))$ is true. Therefore, for all integers $x$ and $y$ the statement ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0)) is true.

Answer: true

(2) $∀x((x < 0) ∨ (x^2 ≥ x))$

Since for each integer $x$ the statement $x^2\ge x$ is true, we conclude the disjunction $(x < 0) ∨ (x^2 ≥ x)$ is also true, and hence the statement $∀x((x < 0) ∨ (x^2 ≥ x))$ is true.

Answer: true

(3) $¬(∃xP(x)) ↔ (∀x¬P(x))$ for all predicates $P(x)$

Since $¬(∃xP(x)) \equiv (∀x¬P(x)),$ the statements $¬(∃xP(x))$ and $(∀x¬P(x))$ have the same values, and hence the equivalence $¬(∃xP(x)) ↔ (∀x¬P(x))$ is true.

Answer: true