Question 63

1. Let $d$ represent a day from the domain of all days.

   • $\text{Rainy}(d)$: "day $d$ is rainy"
   • $\text{Cold}(d)$: "day $d$ is cold"

2. The statement "All rainy days are cold" is formalized as:

   $$\forall d (\text{Rainy}(d) \to \text{Cold}(d)).$$

3. The sentence "Not all rainy days are cold" is the direct logical negation:

   $$\neg \forall d (\text{Rainy}(d) \to \text{Cold}(d)).$$

4. Apply Quantifier and Implication Negation:

   • Negating the universal quantifier yields an existential quantifier:
     $$\exists d \neg (\text{Rainy}(d) \to \text{Cold}(d)).$$
   • Since $p \to q \equiv \neg p \lor q$, its negation is $\neg (p \to q) \equiv p \land \neg q$:
     $$\exists d (\text{Rainy}(d) \land \sim\text{Cold}(d)).$$

This corresponds exactly to option D.