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Let's analyze each proposition to determine which one is true:

**A. \(\exists x \exists y \, (x^2 + y^2 = 25)\)**  
Where the domain of \(x\) and \(y\) is the set of all integers.

- This means there exist integers \(x\) and \(y\) such that \(x^2 + y^2 = 25\).  
- The solutions to \(x^2 + y^2 = 25\) are integer pairs like \((x, y) = (3, 4), (4, 3), (5, 0), (0, 5)\), etc.  
- Since such pairs exist, this proposition is **true**.

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**B. \(\forall a \forall b \exists x \, (ax = b)\)**  
Where the domain of \(a\), \(b\), and \(x\) is the set of all real numbers.

- This proposition states that for every pair of real numbers \(a\) and \(b\), there exists an \(x\) such that \(ax = b\).  
- If \(a \neq 0\), we can solve for \(x = \frac{b}{a}\), which works.  
- However, if \(a = 0\) and \(b \neq 0\), no real \(x\) satisfies \(0 \cdot x = b\). Thus, the proposition does not hold in all cases, making it **false**.

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**C. \(\exists a \exists b \forall x \forall y \, (ax + by = 1)\)**  
Where the domain of \(a\), \(b\), \(x\), and \(y\) is the set of all real numbers.

- This means there exist \(a\) and \(b\) such that for all \(x\) and \(y\), the equation \(ax + by = 1\) holds.  
- This is impossible because \(ax + by = 1\) depends on the values of \(x\) and \(y\); it cannot hold for all possible values of \(x\) and \(y\). Thus, this proposition is **false**.

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**D. The sum of any two irrational numbers is an irrational number.**

- This is false because the sum of two irrational numbers is not always irrational. For example, \(\sqrt{2}\) and \(-\sqrt{2}\) are both irrational, but their sum is 0, which is rational. Hence, this statement is **false**.

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**Conclusion:**  
The only true proposition is **A**.

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Would you like further details on any of these propositions? 

Here are some follow-up questions:

1. Can you find all integer solutions to \(x^2 + y^2 = 25\)?
2. What would happen if the domain of \(a\) and \(b\) in proposition B was restricted to nonzero values?
3. Can you provide other examples where the sum of two irrational numbers results in a rational number?
4. How would proposition C change if it stated \(ax + by = c\), where \(c\) is a constant?
5. In which contexts is the general statement "the sum of irrational numbers is irrational" valid?

**Tip:** When dealing with quantifiers, consider edge cases (like \(a = 0\)) to test the validity of the statement.

Which of the following propositions is true?
A. ∃x∃y, (x² + y² = 25), where the domain of x and y is the set of all integers.
B. ∀a∀b∃x, (ax = b), where the domain of a, b and x is the set of all real numbers.
C. ∃a∃b∀x∀y, (ax + by = 1), where the domain of a, b, x and y is the set of all real numbers.
D. The sum of any two irrational numbers is an irrational number.

Explore which mathematical proposition is true among quantifier-based equations and properties of irrational numbers. Analyze integer solutions to x² + y² = 25 and the behavior of irrational number sums. Suitable for high school students (Grades 10-12).

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