Question 11

Identify the propositional variables:

• $p$: “It is raining”
• $q$: “It is cold”
• $r$: “It is pleasant”

Deconstruct the sentence into its two main conjunctive parts:

1. “It is not raining and it is pleasant”: $\neg p \wedge r$.
2. “and”: represented by the logical conjunction $\wedge$.
3. “it is not pleasant only if it is raining and it is cold”:
   In propositional logic, the conditional construction “$A$ only if $B$” translates to $A \to B$.
   Here, $A$ is “it is not pleasant” ($\neg r$), and $B$ is “it is raining and it is cold” ($p \wedge q$).
   Therefore, “$\neg r$ only if $(p \wedge q)$” translates directly to $\neg r \to (p \wedge q)$.

Combining the two clauses with conjunction yields:

This corresponds to option A.