Let's analyze the logical statement given the new set $U = \mathbb{Z}$ (the set of integers) and the predicate $P(x, y, z)$ defined as:

$P(x, y, z) : x + y - z = xy$

The problem asks us to determine whether the following statement is true:

$\forall z \, \exists y \, P(5, y, z)$

In other words, we need to check whether, for every integer $z$, there exists an integer $y$ such that the equation $5 + y - z = 5y$ holds true.

Step 1: Simplify the equation

Starting with the equation:

$5 + y - z = 5y$

We can rearrange this to solve for $y$:

$5 - z = 5y - y$

$5 - z = 4y$

$y = \frac{5 - z}{4}$

Step 2: Analyze the solution

For $y$ to be an integer, the expression $\frac{5 - z}{4}$ must also be an integer. This means that $5 - z$ must be divisible by 4.

So, the condition becomes:

$5 - z \equiv 0 \pmod{4}$

This can be rewritten as:

$z \equiv 1 \pmod{4}$

Step 3: Determine the truth of the statement

Given the condition $z \equiv 1 \pmod{4}$, this equation only holds when $z$ is congruent to 1 modulo 4. Therefore, $y$ exists and is an integer for such values of $z$.

However, the original statement $\forall z \, \exists y \, P(5, y, z)$ requires that for every integer $z$, there exists a corresponding integer $y$ that satisfies the equation. This is not true because there are values of $z$ that do not satisfy $z \equiv 1 \pmod{4}$, which would make $y$ non-integral.

Conclusion

The statement:

$\forall z \, \exists y \, P(5, y, z)$

is false because there are integers $z$ for which no integer $y$ can satisfy the equation $5 + y - z = 5y$.

Do you have any questions or would you like further details?

Here are some related questions you might consider:

1. What specific values of $z$ fail to satisfy the equation, and why?
2. How would the solution change if $x$ were a different integer?
3. What if the equation were slightly modified, e.g., $x + y - z = xy + 1$?
4. How can you determine for which integers $z$ the equation has an integer solution for $y$?
5. Can you generalize this result to other similar equations involving integers?

Tip: When solving equations over integers, check if the solution requires divisibility by a certain number—this can restrict the values of variables and impact the truth of logical statements.