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The exercise is asking for the negation of the following propositions:

1. **Proposition r:**
   \[
   \forall x \in \mathbb{R}, -1 \leq \frac{2x}{x^2+1} \leq 1.
   \]
   
2. **Proposition p:**
   \[
   \exists (x, y, z) \in \mathbb{R}^3 : \frac{x+1}{7} = \frac{y-5}{2} = \frac{z}{13}.
   \]

3. **Proposition q:**
   \[
   \forall x \in \mathbb{R}, x^2 \in \mathbb{Q} \Rightarrow x \in \mathbb{Q}.
   \]

Now, let's provide the negations for each:

---

### Negation of **r**:
The original statement is:
\[
\forall x \in \mathbb{R}, -1 \leq \frac{2x}{x^2 + 1} \leq 1.
\]
The negation would be:
\[
\exists x \in \mathbb{R} \text{ such that } \left( \frac{2x}{x^2 + 1} < -1 \right) \text{ or } \left( \frac{2x}{x^2 + 1} > 1 \right).
\]
This means there exists at least one \( x \) where the inequality is not satisfied.

---

### Negation of **p**:
The original statement is:
\[
\exists (x, y, z) \in \mathbb{R}^3 : \frac{x+1}{7} = \frac{y-5}{2} = \frac{z}{13}.
\]
The negation would be:
\[
\forall (x, y, z) \in \mathbb{R}^3, \frac{x+1}{7} \neq \frac{y-5}{2} \text{ or } \frac{y-5}{2} \neq \frac{z}{13}.
\]
This means for all possible triples \( (x, y, z) \), the equality does not hold.

---

### Negation of **q**:
The original statement is:
\[
\forall x \in \mathbb{R}, x^2 \in \mathbb{Q} \Rightarrow x \in \mathbb{Q}.
\]
The negation would be:
\[
\exists x \in \mathbb{R} \text{ such that } x^2 \in \mathbb{Q} \text{ and } x \notin \mathbb{Q}.
\]
This means there exists some real number \( x \) such that \( x^2 \) is rational, but \( x \) itself is not rational.

---

Do you want further details or have any questions?

Here are 5 related questions you might explore:
1. How do logical quantifiers affect negation?
2. Can you find an example where \( \frac{2x}{x^2 + 1} \) exceeds the given bounds?
3. What is the geometric meaning of the system of equations in proposition \( p \)?
4. Can you think of a real number whose square is rational but the number itself is not?
5. How does negating a mathematical implication \( P \Rightarrow Q \) differ from negating conjunctions or disjunctions?

**Tip:** To negate a statement involving "for all" (\( \forall \)) or "there exists" (\( \exists \)), always remember that these switch roles during negation: \( \forall \) becomes \( \exists \) and \( \exists \) becomes \( \forall \).

Donner la négation des propositions suivantes :

1. ∀x ∈ ℝ : -1 ≤ (2x / (x^2 + 1)) ≤ 1.
2. ∃(x, y, z) ∈ ℝ³ : (x + 1)/7 = (y - 5)/2 = z/13.
3. ∀x ∈ ℝ : x² ∈ ℚ ⇒ x ∈ ℚ.

Explore the negation of logical propositions involving real and rational numbers. The exercise covers the negation of inequalities, existential quantifiers, and implications with detailed solutions. Suitable for university-level students studying mathematical logic.

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