For each of the following equivalences, determine if it is valid for all predicates P and Q. If valid then give a full explanation. If invalid then provide a counter example. (a) Vx(P(x) A Q(x)) = VxP(x) \¬xQ(x) (b) Vx(¬P(x) V Q(x)) = (¬³xP(x)) V (VxQ(x)) (c) 3x, y (P(x) V Q(y)) = 3x (P(x) V Q(x)) (d) 3x, y (P(x) ^ ¬Q(y)) = ((3x(P(x))) ^ (¬\y (Q(y))))
For each of the following equivalences, determine if it is valid for all predicates P and Q. If valid then give a full explanation. If invalid then provide a counter example. (a) Vx(P(x) A Q(x)) = VxP(x) \¬xQ(x) (b) Vx(¬P(x) V Q(x)) = (¬³xP(x)) V (VxQ(x)) (c) 3x, y (P(x) V Q(y)) = 3x (P(x) V Q(x)) (d) 3x, y (P(x) ^ ¬Q(y)) = ((3x(P(x))) ^ (¬\y (Q(y))))
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Transcribed Image Text:For each of the following equivalences, determine if it
is valid for all predicates P and Q. If valid then give a full explanation. If
invalid then provide a counter example.
(a) Vx(P(x) A Q(x)) = VxP(x) \¬xQ(x)
(b) Vx(¬P(x) V Q(x)) = (¬³xP(x)) V (VxQ(x))
(c) 3x, y (P(x) V Q(y)) = 3x (P(x) V Q(x))
(d) 3x, y (P(x) ^ ¬Q(y)) = ((3x(P(x))) ^ (¬\y (Q(y))))
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