1. The problem asks to express "There are exactly 8 planets in our solar system" using predicate logic, where $P(x)$ means "$x$ is a planet in our solar system". 2. To say "there are exactly 8 planets," we need to express that there exist 8 distinct celestial bodies $x_1, x_2, \ldots, x_8$ such that each is a planet, and any planet is one of these 8. 3. Formally, this can be written as: $$\exists x_1 \exists x_2 \exists x_3 \exists x_4 \exists x_5 \exists x_6 \exists x_7 \exists x_8 \bigg( \bigwedge_{1 \leq i < j \leq 8} x_i \neq x_j \wedge \bigwedge_{i=1}^8 P(x_i) \wedge \forall y (P(y) \rightarrow (y = x_1 \vee y = x_2 \vee y = x_3 \vee y = x_4 \vee y = x_5 \vee y = x_6 \vee y = x_7 \vee y = x_8)) \bigg)$$ 4. Explanation: - $\exists x_1 \ldots x_8$ states there are at least 8 planets. - $x_i \neq x_j$ for $i \neq j$ ensures these 8 are distinct. - $P(x_i)$ for each $i$ means each $x_i$ is a planet. - $\forall y (P(y) \rightarrow (y = x_1 \vee \cdots \vee y = x_8))$ means any planet $y$ must be one of these 8, so there are no more than 8. This fully captures "exactly 8 planets in our solar system."