1. **State the problem:** Construct truth tables for the following statements: a. $\left(\sim q \wedge r\right) \vee \left[p \wedge \left(q \wedge \sim r\right)\right]$ b. $\left(p \to q\right) \to \left(q \vee r\right)$ c. $\left(p \wedge \sim r\right) \leftrightarrow \left(q \vee r\right)$ 2. **Recall truth table basics:** - $p, q, r$ are propositions that can be true (T) or false (F). - $\sim$ means NOT (negation). - $\wedge$ means AND. - $\vee$ means OR. - $\to$ means implication: $p \to q$ is false only if $p$ is true and $q$ is false. - $\leftrightarrow$ means biconditional: true if both sides have the same truth value. 3. **List all possible truth values for $p, q, r$:** There are $2^3=8$ combinations. | $p$ | $q$ | $r$ | |-----|-----|-----| | T | T | T | | T | T | F | | T | F | T | | T | F | F | | F | T | T | | F | T | F | | F | F | T | | F | F | F | 4. **Compute each expression step-by-step:** ### a. $\left(\sim q \wedge r\right) \vee \left[p \wedge \left(q \wedge \sim r\right)\right]$ | $p$ | $q$ | $r$ | $\sim q$ | $\sim r$ | $\sim q \wedge r$ | $q \wedge \sim r$ | $p \wedge (q \wedge \sim r)$ | Final (a) | |-----|-----|-----|----------|----------|-------------------|-------------------|-------------------------------|-----------| | T | T | T | F | F | F | F | F | F | | T | T | F | F | T | F | T | T | T | | T | F | T | T | F | T | F | F | T | | T | F | F | T | T | F | F | F | F | | F | T | T | F | F | F | F | F | F | | F | T | F | F | T | F | T | F | F | | F | F | T | T | F | T | F | F | T | | F | F | F | T | T | F | F | F | F | ### b. $\left(p \to q\right) \to \left(q \vee r\right)$ | $p$ | $q$ | $r$ | $p \to q$ | $q \vee r$ | Final (b) | |-----|-----|-----|-----------|------------|-----------| | T | T | T | T | T | T | | T | T | F | T | T | T | | T | F | T | F | T | T | | T | F | F | F | F | F | | F | T | T | T | T | T | | F | T | F | T | T | T | | F | F | T | T | T | T | | F | F | F | T | F | F | ### c. $\left(p \wedge \sim r\right) \leftrightarrow \left(q \vee r\right)$ | $p$ | $q$ | $r$ | $\sim r$ | $p \wedge \sim r$ | $q \vee r$ | Final (c) | |-----|-----|-----|----------|--------------------|------------|-----------| | T | T | T | F | F | T | F | | T | T | F | T | T | T | T | | T | F | T | F | F | F | T | | T | F | F | T | T | F | F | | F | T | T | F | F | T | F | | F | T | F | T | F | T | F | | F | F | T | F | F | F | T | | F | F | F | T | F | F | T | 5. **Summary:** - Expression (a) is true when either $\sim q \wedge r$ or $p \wedge (q \wedge \sim r)$ is true. - Expression (b) is false only when $p \to q$ is true and $q \vee r$ is false. - Expression (c) is true when $p \wedge \sim r$ and $q \vee r$ have the same truth value. This completes the truth tables for the given statements.