1. **Problem 1:** Given the predicate $P(x,y): xy = 1$ with $x,y \in \mathbb{R}$, evaluate the truth value of the statement $\forall x \in \mathbb{R}, \exists y \in \mathbb{R}$ such that $P(x,y)$ holds. 2. For each real number $x$, we want to find a real number $y$ such that $xy = 1$. 3. If $x \neq 0$, then $y = \frac{1}{x}$ satisfies $xy = 1$. 4. However, if $x = 0$, there is no real $y$ such that $0 \cdot y = 1$ because $0 \neq 1$. 5. Therefore, the statement $\forall x \in \mathbb{R}, \exists y \in \mathbb{R}$ with $xy=1$ is **false** because it fails at $x=0$. 6. **Problem 2:** Given the predicate $P(x,y): xy \geq 0$ with $x,y \in \mathbb{Z}$, evaluate the truth value of the statement $\exists x \in \mathbb{Z}, \forall y \in \mathbb{Z}$ such that $P(x,y)$ holds. 7. We want to find an integer $x$ such that for every integer $y$, $xy \geq 0$. 8. Consider $x=0$. For any integer $y$, $0 \cdot y = 0 \geq 0$. 9. Thus, $x=0$ satisfies $\forall y \in \mathbb{Z}, xy \geq 0$. 10. Therefore, the statement $\exists x \in \mathbb{Z}, \forall y \in \mathbb{Z}$ with $xy \geq 0$ is **true**. **Final answers:** - Statement 1 is **false**. - Statement 2 is **true**.