GATE CS 1998 Q1.5 — Propositional and First Order Logic

To find the converse of the given assertion, let us first translate it into propositional logic.

Let the propositions be defined as:

• $P$: "I stay"
• $Q$: "You go"

The original statement is: "I stay only if you go".

In propositional logic, the phrase "$P$ only if $Q$" is equivalent to the implication:

The converse of an implication $P \rightarrow Q$ is defined as:

Translating $Q \rightarrow P$ back into English gives:

• "If you go, then I stay", which is equivalent to "I stay if you go".

Let's analyze the given options:

• A. I stay if you go: This is $Q \rightarrow P$ (Converse). Correct.
• B. If I stay then you go: This is $P \rightarrow Q$ (Original statement).
• C. If you do not go then I do not stay: This is $\neg Q \rightarrow \neg P$ (Contrapositive).
• D. If I do not stay then you go: This is $\neg P \rightarrow Q$.