Give a proof or counterexample, whichever is appropriate. 1. NOT (∃x, (P(x) OR Q(x) OR R(x)))...

Give a proof or counterexample, whichever is appropriate.

1. NOT (∃x, (P(x) OR Q(x) OR R(x))) is logically equivalent to ∀x, ((NOT P(x)) AND (NOT Q(x)) AND (NOT R(x))).

2. NOT (∃x, (P(x) AND Q(x) AND R(x))) is logically equivalent to ∀x, ((NOT P(x)) OR (NOT Q(x)) OR (NOT R(x))).

3. NOT (∃x, (P(x) ⇒ Q(x))) is logically equivalent to ∀x, (P(x)⇒ NOT Q(x)).

4. NOT (∃x, (P(x) ⇒ Q(x))) is logically equivalent to ∀x, (P(x) AND (NOT Q(x))).

5. ∃x, (P(x) OR Q(x)) is logically equivalent to (∃x, P(x)) OR (∃x, Q(x)).

6. ∃x, (P(x) AND Q(x)) is logically equivalent to (∃x, P(x)) AND (∃x, Q(x)).

7. ∀x, (P(x) OR Q(x)) is logically equivalent to (∀x, P(x)) OR (∀x, Q(x)).

8. ∀x, (P(x) AND Q(x)) is logically equivalent to (∀x, P(x)) AND (∀x, Q(x)).

9. ∀x, (P (x) ⇒ Q(x)) is logically equivalent to (∀x, (x)) ⇒ (∀x, Q(x)).

Expert Solution

Colby Messinger answered 2 years ago

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