Let P(x) and Q(x) be predicates defined on the same domain D. For each pair of s

Question

Let P(x) and Q(x) be predicates defined on the same domain D. For each pair of statement forms below, explain whether or not they are logically equivalent. If not, why not? If they are equivalent, explain why. (a) Is Forall x elementof D, (P(x) Lambda Q(x)) equivalent to (Forall x elementof D, P(x)) Lambda (Forall x elementof D, Q(x))? (b) Is exist x elementof D, (P(x) Lambda Q(x)) equivalent to (exist x elementof D, P(x)) Lambda (exist x elementof D, Q(x))? (c) Is Forall x elementof D, (P(x) Q(x)) equivalent to (Forall x elementof D, P(x)) (Forall x elementof D, Q(x))? (d) Is exist x elementof D, (P(x) Q(x)) equivalent to (exist x elementof D, P(x)) (Forall x elementof D, Q(x))?

Explanation / Answer

A is true as we already have a theorem as follows:

x[P(x) Q(x)] (xP(x) xQ(x))

B is not true as per the theorem

x[P(x) Q(x)] is not (xP(x) xQ(x))

c is not true as per the theorem x[P(x) Q(x)] is not (xP(x) xQ(x))

d is true as per the theorem x[P(x) Q(x)] (xP(x) xQ(x))

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