Question

If P and Q are predicates over some domain, and if it is true that Vx(P(x)V Q(x)), must VxP(x) v VæQ(x) also be true? Explain.

Answer 1

It is not true

Explanation:

Suppose your domain is the integer numbers. Define

P(x)="x is even"

Q(x)="x is odd"

So we have that the predicate
`\forall x(P(x) \vee Q(x))` is always true because the integers are always even or odd. But the predicate
`\forall x P(x) \vee \forall x Q(x)` means that all the integer numbers are even or all the integer numbers are odd, which is false. So we can't deduce
`\forall x P(x) \vee \forall x Q(x)` from
`\forall x(P(x) \vee Q(x))`.