The CORRECT formula for the sentence, “not all rainy days are cold” is

To translate the English statement into first-order logic:

1. Let $\text{Rainy}(d)$ mean "$d$ is a rainy day" and $\text{Cold}(d)$ mean "$d$ is a cold day".

2. "All rainy days are cold" is represented as:

   $$\forall d\ (\text{Rainy}(d) \rightarrow \text{Cold}(d))$$

3. "Not all rainy days are cold" is the negation of this statement: $$

eg \forall d\ ($\text{Rainy}$(d) \rightarrow $\text{Cold}$(d))$$

4. Using quantifier negation $

eg \forall d\ \phi \equiv \exists d\ eg \phi$:$$\exists d\ eg(\text{Rainy}(d) \rightarrow \text{Cold}(d))$$

5. Applying the implication equivalence $

eg(P \rightarrow Q) \equiv P \land eg Q$:

Thus, option D is the correct formula.