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#75
by
mikoo2ww22
- opened
😃: Show that ∀xP (x) ∨ ∀xQ(x) and ∀x(P (x) ∨ Q(x)) are
not logically equivalent.
🤖: To show that ∀xP(x) ∨ ∀xQ(x) and ∀x(P(x) ∨ Q(x)) are not logically equivalent, we need to find a pair of statements that satisfy one but not the other.
Let's consider two possible pairs of statements:
- P = "The sky is blue" and Q = "The grass is green."
- P = "Aardvarks have four legs" and Q = "Giraffes have necks longer than any other animal."
In both cases, we can easily see that:
- ∀xP(x) ∨ ∀xQ(x): T → F (true in both cases)
- ∀x(P(x) ∨ Q(x)): F → T (false in case 1, true in case 2)
So, while ∀xP(x) ∨ ∀xQ(x) and ∀x(P(x) ∨ Q(x)) are related, they are not logically equivalent.