The CORRECT formula for the sentence, "not all Rainy days are Cold" is

Key idea: "Not all Rainy days are Cold" means there is at least one day that is Rainy and not Cold.

• Start from the universal statement: "All Rainy days are Cold" is expressed as ∀d (Rainy(d) → Cold(d)).

• Negate it: "Not all Rainy days are Cold" is ¬∀d (Rainy(d) → Cold(d)), which is equivalent to ∃d ¬(Rainy(d) → Cold(d)).

• Use the equivalence ¬(P → Q) ≡ P ∧ ¬Q to get ∃d (Rainy(d) ∧ ¬Cold(d)).

• Therefore the correct formalization is: ∃d (Rainy(d) ∧ ¬Cold(d)).

Why the other given formulas are incorrect:

• ∀d (Rainy(d) ∧ ¬Cold(d)) incorrectly claims every day is rainy and not cold, which is much stronger than "not all".

• ∀d (¬Rainy(d) → Cold(d)) says for every day, if it is not rainy then it is cold (equivalently every day is Rainy or Cold). This does not assert existence of a rainy day that is not cold.

• ∃d (¬Rainy(d) → Cold(d)) only asserts there is some day where the implication holds; because implications can be true when the antecedent is false, this does not require a rainy day that is not cold.

Final answer: ∃d (Rainy(d) ∧ ¬Cold(d)).