I absolutely STUMPED on this one

Ok guys. Here's a brain-drainer. Show:

[For every x P(x)] OR [For every x Q(x)]

and

For every[ P(x) OR Q(x) ]

are NOT logically equivilent.

This is really giving me sleep problems. I can get this to work with AND (instead of OR). You simply Let P(x) = "x is even" and Q(x) = "x is odd" so therefor, the first is true, but the second is not (because a number can't be even AND odd).

So how do you do it for OR?

just some clarification:

1) Is there any difference in the question between inclusive and exclusive disjunction (i.e. is OR different to XOR)?? Is your OR mean and/or, or 'OR and not AND'

2) should part 2 be : "For every x [P(x) OR Q(x)]"

3) I must be misunderstanding you, cos i get, with and, that the first is FALSE, and the 2nd is FALSE as well.
i.e. Not all x are odd, not all x are even. ALSO, not all x are even and odd.


If this is what i'm thinking, then you are close. stick with P(x) = 'x is even' and Q(x) = 'x is odd'

then you get:
a) [For every x, P(x)] = false {not all numbers are even}
b) [For every x, Q(x)] = false {not all numbers are odd}
=> The first part is False OR False = False

c) [For every x, P(x) OR Q(x)] = true {all numbers are even or odd}
=> The 2nd is true

=> the 1st isn't the same as the 2nd...