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.