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The image you provided contains a set of mathematical propositions labeled 25 to 32. Below is an analysis and explanation for each statement. I'll evaluate their truth value (whether the statement is true or false) and provide the negation (counter-statement) for each:

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### **25. \( A = \forall x \in \mathbb{R}, \ |x| = -x^3 \)**
- **Truth**: False. For real numbers, \( |x| \) (the absolute value of \( x \)) is non-negative, while \( -x^3 \) is negative for positive \( x \), which contradicts the equality for \( x > 0 \).
- **Negation**: \( \exists x \in \mathbb{R} \) such that \( |x| \neq -x^3 \).

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### **26. \( B = \exists x \in \mathbb{Q}, \ x^2 - 2x > -2 \)**
- **Truth**: True. For example, if \( x = 1 \in \mathbb{Q} \), we have \( 1^2 - 2(1) = -1 \), which is greater than \( -2 \).
- **Negation**: \( \forall x \in \mathbb{Q}, \ x^2 - 2x \leq -2 \).

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### **27. \( C = \forall n \in \mathbb{N}, \ 4 : n^2 \rightarrow 4 : n \)**
- **Interpretation**: This looks like a divisibility statement. It seems to mean that for all natural numbers \( n \), if 4 divides \( n^2 \), then 4 also divides \( n \).
- **Truth**: True. If \( 4 \) divides \( n^2 \), then \( n \) must be divisible by 2, and hence \( n^2 \) is divisible by \( 4 \).
- **Negation**: \( \exists n \in \mathbb{N} \) such that \( 4 \mid n^2 \) but \( 4 \nmid n \).

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### **28. \( D = \exists x \in \mathbb{R}, \ \sin x + 2x = 1 \)**
- **Truth**: True. There exists an \( x \in \mathbb{R} \) for which \( \sin x + 2x = 1 \), though finding the exact value might require numerical methods.
- **Negation**: \( \forall x \in \mathbb{R}, \ \sin x + 2x \neq 1 \).

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### **29. \( E = \forall x \in \mathbb{R}, \ \frac{x+1}{x} > 2 \)**
- **Truth**: False. For instance, if \( x = 1 \), the expression evaluates to \( \frac{1+1}{1} = 2 \), which is not greater than 2.
- **Negation**: \( \exists x \in \mathbb{R}, \ \frac{x+1}{x} \leq 2 \).

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### **30. \( F = \exists x \in \mathbb{Q}, \ x^2 + 1 = 0 \)**
- **Truth**: False. The equation \( x^2 + 1 = 0 \) has no solutions in \( \mathbb{Q} \) because it implies \( x^2 = -1 \), and there are no rational numbers whose square is negative.
- **Negation**: \( \forall x \in \mathbb{Q}, \ x^2 + 1 \neq 0 \).

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### **31. \( G = \forall x \in \mathbb{N}, \ (x^3 : 3) \rightarrow (x : 4) \)**
- **Interpretation**: This might mean that for all natural numbers \( x \), if 3 divides \( x^3 \), then 4 divides \( x \).
- **Truth**: False. For example, if \( x = 3 \), \( 3 \mid 3^3 \), but \( 4 \nmid 3 \).
- **Negation**: \( \exists x \in \mathbb{N}, \ 3 \mid x^3 \) but \( 4 \nmid x \).

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### **32. \( H = \forall x \in \mathbb{N}, \ (x^2 + x) : 2 \)**
- **Truth**: True. For all natural numbers \( x \), \( x^2 + x = x(x+1) \) is always even because one of \( x \) or \( x+1 \) is even.
- **Negation**: \( \exists x \in \mathbb{N}, \ x^2 + x \) is not divisible by 2.

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Would you like more details on any of these statements or further clarification?

Here are 5 related questions to expand this topic:
1. What is the general approach to negating a mathematical proposition?
2. How can you solve trigonometric equations involving both sine and linear terms like in proposition 28?
3. What are the key properties of divisibility in natural numbers, as seen in propositions 27 and 31?
4. How can numerical methods be applied to solve transcendental equations?
5. Why does the equation \( x^2 + 1 = 0 \) have no real or rational solutions?

**Tip**: When analyzing mathematical propositions, always double-check the domain of the variables (e.g., real numbers, rational numbers) to ensure the correct interpretation of the statements.

The user submitted a query containing several mathematical propositions to be evaluated and negated, ranging from absolute values to divisibility and trigonometric equations.

This page covers the detailed analysis and negation of a set of mathematical propositions involving absolute values, divisibility, and trigonometric equations. Suitable for undergraduate students, this analysis includes key concepts such as the divisibility theorem, properties of absolute values, and solutions to trigonometric equations like sin(x) + 2x = 1.

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