GATE CS 2014 Set 3 Q1 — Propositional and First Order Logic

Let us represent the statements using predicates. Let $x$ represent a mobile phone, and define the following predicates:

• $G(x)$: $x$ is a good mobile phone
• $C(x)$: $x$ is a cheap mobile phone

Now, let's translate the given statements $P$ and $Q$ into logical expressions:

• Statement P: "Good mobile phones are not cheap"
  $$\forall x \, (G(x) \implies \neg C(x))$$
• Statement Q: "Cheap mobile phones are not good"
  $$\forall x \, (C(x) \implies \neg G(x))$$

By the law of contraposition, any implication $A \implies B$ is logically equivalent to its contrapositive $\neg B \implies \neg A$.

Applying contraposition to the implication in statement $P$:

Since this is exactly the expression for statement $Q$, the statements $P$ and $Q$ are logically equivalent ($P \equiv Q$).

Because $P$ and $Q$ are logically equivalent:

1. $P \implies Q$ is TRUE (so $L$ is TRUE).
2. $Q \implies P$ is TRUE (so $M$ is TRUE).
3. $P$ is equivalent to $Q$ is TRUE (so $N$ is TRUE).

Therefore, $L$, $M$, and $N$ are all TRUE.