Question

Why aren't these two quantified statements logically equivalent? ∃x(P(x) xor Q(x)) and (∃x(P(x)) xor (∃xQ(x))?

Answer 1

I thought you could use distributive law in the first one to get the second but its asking to show that they aren't logically equivalent..

Answer 2

The first statement says that there exists and x such that for that single x, exactly one of P or Q is true.

The second statement is extremely limiting, such that if *any* x exists for which P is true, the there is no x for which Q is true. Where the first statement only says that there exists one x for which P is true and Q is false (or Q true P false).

Answer 3

Is there any way to show it mathematically?

Answer 4

A counter example, for the set of integers:
P(x) is true if a number is odd
Q(x) is true if a number is even
The first statement is true for any integer x, since P(x) xor Q(x) is always true.

For that same set of numbers
∃x(P(x)) is true
∃x(Q(x)) is true
P xor Q is then false.