Truth Table D64979

1. **State the problem:** Construct the truth table for the logical expression $$ (r \wedge q) \lor (p \wedge q) $$ where $\wedge$ means AND and $\lor$ means OR. 2. **Recall the rules:** - $p \wedge q$ is true only if both $p$ and $q$ are true. - $p \lor q$ is true if at least one of $p$ or $q$ is true. 3. **Evaluate each part:** - Compute $r \wedge q$ for each row. - Compute $p \wedge q$ for each row. - Then compute $(r \wedge q) \lor (p \wedge q)$. 4. **Fill the truth table:** | $p$ | $q$ | $r$ | $r \wedge q$ | $p \wedge q$ | $(r \wedge q) \lor (p \wedge q)$ | |---|---|---|---|---|---| | T | T | T | T | T | T | | T | T | F | F | T | T | | T | F | T | F | F | F | | T | F | F | F | F | F | | F | T | T | T | F | T | | F | T | F | F | F | F | | F | F | T | F | F | F | | F | F | F | F | F | F | 5. **Explanation:** - For example, in the first row, $r=T$, $q=T$, so $r \wedge q = T$; also $p=T$, $q=T$, so $p \wedge q = T$; thus the OR is $T \lor T = T$. - In the third row, $q=F$, so both $r \wedge q$ and $p \wedge q$ are false, so the OR is false. **Final truth table:** | $p$ | $q$ | $r$ | $(r \wedge q) \lor (p \wedge q)$ | |---|---|---|---| | T | T | T | T | | T | T | F | T | | T | F | T | F | | T | F | F | F | | F | T | T | T | | F | T | F | F | | F | F | T | F | | F | F | F | F |