GATE CS 2013 Q27 — Propositional and First Order Logic

Let us define the predicates:

• $F(x)$: $x$ is my friend
• $P(x)$: $x$ is perfect

The statement "None of my friends are perfect" means that if anyone is my friend, they are not perfect. This can be written in first-order logic as:

$$\forall x (F(x) \implies \neg P(x))$$

We can simplify this expression using logical equivalences:

1. Using the implication rule ($A \implies B \equiv \neg A \lor B$):
   $$\forall x (\neg F(x) \lor \neg P(x))$$
2. Using the duality between universal and existential quantifiers ($\forall x \, \phi(x) \equiv \neg \exists x \, \neg \phi(x)$):
   $$\neg \exists x \, \neg (\neg F(x) \lor \neg P(x))$$
3. Applying De Morgan's Law to the inner expression:
   $$\neg \exists x (F(x) \land P(x))$$

This matches Option D, which states that there does not exist any $x$ who is both my friend and perfect.