Question

Determine whether each argument is valid. If the argument is valid, give a proof using the laws of logic. If the argument is invalid, give values for the predicates P and Q over the domain {a, b} that demonstrate the argument is invalid.

(a) ∃x (P(x) ∧ Q(x)) ∴ ∃x Q(x) ∧ ∃x P(x)
(b) ∀x (P(x) ∨ Q(x)) ∴ ∀x Q(x) ∨ ∀x P(x)

Answer 1

The arguments given are invalid because counterexamples can be provided that satisfy the premises while not satisfying the conclusions.

Step-by-step explanation:

The given argument is: ∃x (P(x) ∧ Q(x)) ∴ ∃x Q(x) ∧ ∃x P(x)

To determine whether this argument is valid or not, we can use a counterexample. If we consider the domain {a, b} and let P(a) be true, Q(a) be true, and Q(b) be false, the premise ∃x (P(x) ∧ Q(x)) is satisfied. However, the conclusion ∃x Q(x) ∧ ∃x P(x) is not satisfied, as Q(b) is false. Therefore, the argument is invalid.

The given argument is: ∀x (P(x) ∨ Q(x)) ∴ ∀x Q(x) ∨ ∀x P(x)

To determine whether this argument is valid or not, we can use a counterexample. If we consider the domain {a, b} and let P(a) be true, Q(b) be true, and Q(a) be false, the premise ∀x (P(x) ∨ Q(x)) is satisfied. However, the conclusion ∀x Q(x) ∨ ∀x P(x) is not satisfied, as P(b) is false. Therefore, the argument is also invalid.