1. **State the problem:** We have logical propositions: - p: The sun is shining - q: It is raining - r: The ground is wet We need to make compound statements and truth tables for these: - Column 1: $p \wedge \neg r$ and $p \lor q$ and $r \to p$ - Column 2: $r \leftrightarrow q$ and $r \to (q \wedge \neg p)$ and $p = (\neg r \wedge \neg q)$ 2. **Assign truth values:** With three variables $p$, $q$, $r$, there are $2^3 = 8$ combinations of truth values. 3. **Build truth table:** We list values of $p$, $q$, and $r$ and compute each compound statement. | p | q | r | $p \wedge \neg r$ | $p \lor q$ | $r \to p$ | $r \leftrightarrow q$ | $r \to (q \wedge \neg p)$ | $p = (\neg r \wedge \neg q)$ | |---|---|---|------------------|-------------|----------|-------------------|------------------------|-------------------------------| | T | T | T | F | T | T | T | F | F | | T | T | F | T | T | T | F | T | F | | T | F | T | F | T | T | F | F | F | | T | F | F | T | T | T | F | F | T | | F | T | T | F | T | F | T | T | F | | F | T | F | F | T | T | F | T | F | | F | F | T | F | F | F | T | F | F | | F | F | F | F | F | T | F | F | T | 4. **Explanation:** - $p \wedge \neg r$: True when $p$ is true and $r$ is false. - $p \lor q$: True when at least one of $p$ or $q$ is true. - $r \to p$: True unless $r$ is true and $p$ is false. - $r \leftrightarrow q$: True when $r$ and $q$ have the same truth value. - $r \to (q \wedge \neg p)$: True unless $r$ is true but $(q \wedge \neg p)$ is false. - $p = (\neg r \wedge \neg q)$: True when $p$ equals $(\neg r \wedge \neg q)$. Final answer: The truth table above shows all combinations and values of the compound statements.