1. The first statement is \(P(x,y): xy=1\) with \(x,y \in \mathbb{R}\). The quantified statement is \(\forall x \in \mathbb{R}, \exists y \in \mathbb{R} \text{ such that } xy=1\). For any real \(x \neq 0\), we can find \(y=\frac{1}{x}\) so that \(xy=1\). However, if \(x=0\), there is no real \(y\) satisfying \(0 \cdot y=1\). Therefore, the statement is false because it fails at \(x=0\). 2. The second statement is \(P(x,y): xy \geq 0\) with \(x,y \in \mathbb{Z}\). The quantified statement is \(\exists x \in \mathbb{Z}, \forall y \in \mathbb{Z}, xy \geq 0\). We want to find an integer \(x\) such that for every integer \(y\), the product \(xy\) is nonnegative. - If \(x=0\), then \(xy=0\) for all \(y\), which is \(\geq 0\). - If \(x>0\), then \(xy\) can be negative if \(y<0\). - If \(x<0\), then \(xy\) can be negative if \(y>0\). Thus, \(x=0\) satisfies the condition, making the statement true. Final answers: - First statement: false - Second statement: true