Let P(x, y) be the statement “x
2 + y
2 = 2024.” If the domain for both variables
consists of all real numbers, what are the truth values?
(a) ∃x P(x, √
2024) (b) ∀x ∃y P(x, y) (c) ∃x ∃y P(x, y) (d) ∀x ∃y P(x, y)

Let's analyze the truth values of the statements involving \( P(x, y) \), where \( P(x, y) \) is defined as \( x^2 + y^2 = 2024 \), and the domain for both variables consists of all real numbers.

### (a) \( \exists x \, P(x, \sqrt{2024}) \)

This statement means "There exists an \( x \) such that \( x^2 + (\sqrt{2024})^2 = 2024 \)." 
Since \( (\sqrt{2024})^2 = 2024 \), this simplifies to:
\[
x^2 + 2024 = 2024
\]
\[
x^2 = 0
\]
Thus, \( x = 0 \) is the only solution. Therefore, there exists such an \( x \), so the statement is **true**.

### (b) \( \forall x \, \exists y \, P(x, y) \)

This statement means "For all \( x \), there exists a \( y \) such that \( x^2 + y^2 = 2024 \)."

Rearranging, we need \( y^2 = 2024 - x^2 \). This requires that \( 2024 - x^2 \geq 0 \), i.e., \( x^2 \leq 2024 \), which is always true for all real \( x \) because for any real number, we can find a corresponding \( y \) such that \( y^2 = 2024 - x^2 \). Hence, the statement is **true**.

### (c) \( \exists x \, \exists y \, P(x, y) \)

This statement means "There exists an \( x \) and a \( y \) such that \( x^2 + y^2 = 2024 \)."

For example, if \( x = 0 \) and \( y = \sqrt{2024} \), then \( 0^2 + (\sqrt{2024})^2 = 2024 \). Therefore, the statement is **true**.

### (d) \( \forall x \, \exists y \, P(x, y) \)

This is the same as statement (b). It means "For all \( x \), there exists a \( y \) such that \( x^2 + y^2 = 2024 \)." As shown in the analysis for (b), this is **true**.

### Summary of truth values:
- (a) True
- (b) True
- (c) True
- (d) True

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. What is the geometric interpretation of \( P(x, y) \)?
2. How would the truth values change if the domain were restricted to integers?
3. Can you generalize these statements for \( x^2 + y^2 = r^2 \)?
4. What is the set of all points \( (x, y) \) that satisfy \( x^2 + y^2 = 2024 \)?
5. How does the solution change if \( P(x, y) = x^2 + y^2 \geq 2024 \)?

**Tip:** When working with statements like \( \exists \) (there exists) and \( \forall \) (for all), always carefully consider the domain of the variables involved.

Let P(x, y) be the statement 'x^2 + y^2 = 2024.' If the domain for both variables consists of all real numbers, what are the truth values?
(a) ∃x P(x, √2024)
(b) ∀x ∃y P(x, y)
(c) ∃x ∃y P(x, y)
(d) ∀x ∃y P(x, y)

Explore the truth values of logical statements involving the circle equation x^2 + y^2 = 2024. Learn how to apply algebra and geometric concepts with existential and universal quantifiers to solve each part. Suitable for students in Grades 10-12.

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