U=ℕ

P(x,y,z) : xy-z≤x²-5

∀x∃y∀zP(x,y,z) → ∀zP(5,4,z)

Let's analyze the statement and see if it's true. The problem can be broken down as follows:

Given:
- \( U = \mathbb{N} \) (the set of natural numbers).
- \( P(x, y, z) \) is the predicate \( xy - z \leq x^2 - 5 \).

We want to verify if the implication is true:

\[
\forall x \, \exists y \, \forall z \, P(x, y, z) \rightarrow \forall z \, P(5, 4, z)
\]

### Step 1: Analyze \( \forall x \, \exists y \, \forall z \, P(x, y, z) \)
This means that for every natural number \( x \), there exists a natural number \( y \) such that for all natural numbers \( z \), the predicate \( xy - z \leq x^2 - 5 \) holds true.

Let's express this for arbitrary \( x \):

\[
xy - z \leq x^2 - 5 \quad \text{for all } z
\]

This inequality can be rearranged as:

\[
xy \leq x^2 - 5 + z
\]

### Step 2: Analyze \( \forall z \, P(5, 4, z) \)
Substituting \( x = 5 \) and \( y = 4 \) into the predicate, we get:

\[
5 \cdot 4 - z \leq 5^2 - 5
\]

Simplifying, this becomes:

\[
20 - z \leq 25 - 5
\]

\[
20 - z \leq 20
\]

This simplifies further to:

\[
-z \leq 0 \quad \text{or} \quad z \geq 0
\]

Since \( z \) is a natural number (\( z \in \mathbb{N} \)), this inequality is always true.

### Step 3: Determine if the implication holds
Now, let's review the original implication:

\[
\forall x \, \exists y \, \forall z \, P(x, y, z) \rightarrow \forall z \, P(5, 4, z)
\]

From Step 2, we established that \( \forall z \, P(5, 4, z) \) holds true. Since the right side of the implication is always true, the implication as a whole is also true, regardless of the truth value of the left side.

### Conclusion
The statement:

\[
\forall x \, \exists y \, \forall z \, P(x, y, z) \rightarrow \forall z \, P(5, 4, z)
\]

is **true**.

Would you like more details on any part of the explanation? 

Here are some related questions you might consider:
1. What if \( U = \mathbb{Z} \) instead of \( U = \mathbb{N} \)?
2. How would the analysis change if \( P(x, y, z) \) was a different predicate?
3. Can we find explicit values of \( y \) that satisfy \( \forall x \, \exists y \, \forall z \, P(x, y, z) \)?
4. How does the truth value change if \( x = 0 \) is included in \( U \)?
5. What would happen if \( \exists y \) and \( \forall z \) were swapped in the original quantifier expression?

**Tip:** When dealing with logical implications, if the consequent (right side of the implication) is always true, the entire implication will also always be true, regardless of the antecedent (left side).

Explore the implications of predicate logic and quantifiers in mathematical reasoning. This problem involves analyzing the truth of logical statements using natural numbers. Learn how to assess logical implications step-by-step with detailed examples and solutions.

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