GATE CS 2014 Set 3 Q53 — Propositional and First Order Logic

To translate the sentence "not all Rainy days are Cold" into first-order logic, we can break it down step-by-step:

1. First, represent the statement "all Rainy days are Cold": $$\forall d (\text{Rainy}(d) \to \text{Cold}(d))$$
2. Now, negate this statement to represent "not all Rainy days are Cold": $$\neg \forall d (\text{Rainy}(d) \to \text{Cold}(d))$$
3. Using the quantifier equivalence $\neg \forall x P(x) \equiv \exists x \neg P(x)$, we get: $$\exists d \neg (\text{Rainy}(d) \to \text{Cold}(d))$$
4. Using the implication equivalence $P \to Q \equiv \neg P \vee Q$ and De Morgan's Law, we simplify the inner negation: $$\neg (\text{Rainy}(d) \to \text{Cold}(d)) \equiv \neg (\neg \text{Rainy}(d) \vee \text{Cold}(d)) \equiv \text{Rainy}(d) \wedge \neg \text{Cold}(d)$$

Substituting this back, we get:

This matches Option D.