GATE CS 2017 Set 2 Q11 — Propositional and First Order Logic

To translate the given English statement into a propositional logic formula, we can break it down into smaller components based on the atomic propositions:

• $p$: "It is raining"
• $q$: "It is cold"
• $r$: "It is pleasant"

The given sentence is:
"It is not raining and it is pleasant, and it is not pleasant only if it is raining and it is cold"

We can split this sentence at the central "and" into two main parts:

1. First Part: "It is not raining and it is pleasant"
   • "It is not raining" $\equiv \neg p$
   • "it is pleasant" $\equiv r$
   • Combining them with "and": $(\neg p \wedge r)$
2. Second Part: "it is not pleasant only if it is raining and it is cold"
   • Recall the logical translation for "$A$ only if $B$" is $A \rightarrow B$.
   • Here, $A$ is "it is not pleasant" $\equiv \neg r$.
   • $B$ is "it is raining and it is cold" $\equiv (p \wedge q)$.
   • Thus, "$A$ only if $B$" translates to: $\neg r \rightarrow (p \wedge q)$.

Combining the two parts with the central "and" ($\wedge$), we get:

This matches Option A.