Determining the truth value of this quantified statement

[opticaltempest]

Homework Statement

I am trying to determine the truth value of this statement:

$$\forall \mbox{ }x \mbox{ and } \forall \mbox{ }y, \exists \mbox{ }z \mbox{ such that } y-z=x.$$

Homework Equations

N/A

The Attempt at a Solution

Here is how I arrived at my answer of false. The textbook lists the correct answer as true.

Consider x=0 and y=1.

We have,

1-z=0.

For this statement to be true, z must be equal to 1.

Now, leaving z=1, switch to x=2 and y=0. This gives

0-1=2. (1)

Equation (1) is a false statement. Therefore, there exists no z such that for all x and for all y, y-z=x.

Where is my logic wrong?

Thanks

 

[bdeln]

Hi opticaltempest,

I believe the problem is that you're fixing z, then fixing x and y. The statement is, given some x and y, there exists z.

If you fix all three, then it's easy to find a contradiction to the statement for all x,y, z, we have y - z = x, but this isn't what your original statement said.

Hope that helps!