1. The problem is about understanding the logical conjunction (AND) operation between two propositions $p$ and $q$. 2. The conjunction $p \wedge q$ is true only when both $p$ and $q$ are true. 3. The problem also explains related expressions: - "p but q" means $p \wedge q$ (both $p$ and $q$ are true). - "neither p nor q" means $\neg p \wedge \neg q$ (both $p$ and $q$ are false). 4. Given examples: - "It is hot not but it is sunny" translates to $S \wedge \neg H$ (It is sunny and not hot). - "It is neither hot nor sunny" translates to $\neg H \wedge \neg S$ (It is not hot and not sunny). 5. The truth table for $p \wedge q$ is: $$ \begin{array}{c|c|c} p & q & p \wedge q \\ \hline T & T & T \\ T & F & F \\ F & T & F \\ F & F & F \\ \end{array} $$ 6. To solve problems involving conjunctions, identify the truth values of $p$ and $q$ and apply the rule that $p \wedge q$ is true only if both are true. Final answer: This is a logical conjunction operation in propositional logic, and it is solved by evaluating the truth values of $p$ and $q$ according to the truth table above.