1. **State the problem:** Construct truth tables for the following statements: a. $(\sim q \wedge r) \vee [p \wedge (q \wedge \sim r)]$ b. $(p \to q) \to (q \vee r)$ c. $(p \wedge \sim r) \leftrightarrow (q \vee r)$ 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 (true if both operands are true). - $\vee$ means OR (true if at least one operand is true). - $\to$ means implication (false only if antecedent true and consequent 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. $(\sim q \wedge r) \vee [p \wedge (q \wedge \sim r)]$ - Compute $\sim q$ - Compute $q \wedge \sim r$ - Compute $p \wedge (q \wedge \sim r)$ - Compute $\sim q \wedge r$ - Finally, compute the OR of the two parts b. $(p \to q) \to (q \vee r)$ - Compute $p \to q$ - Compute $q \vee r$ - Compute implication from $p \to q$ to $q \vee r$ c. $(p \wedge \sim r) \leftrightarrow (q \vee r)$ - Compute $\sim r$ - Compute $p \wedge \sim r$ - Compute $q \vee r$ - Compute biconditional 5. **Truth tables:** | $p$ | $q$ | $r$ | $\sim q$ | $\sim r$ | $q \wedge \sim r$ | $p \wedge (q \wedge \sim r)$ | $\sim q \wedge r$ | a: $(\sim q \wedge r) \vee [p \wedge (q \wedge \sim r)]$ | $p \to q$ | $q \vee r$ | b: $(p \to q) \to (q \vee r)$ | $p \wedge \sim r$ | c: $(p \wedge \sim r) \leftrightarrow (q \vee r)$ | |-----|-----|-----|----------|----------|--------------------|-------------------------------|------------------|------------------------------------------------------------|----------|----------|------------------------------|----------------|---------------------------------------------| | T | T | T | F | F | F | F | F | F | T | T | T | F | F | | T | T | F | F | T | T | T | F | T | T | T | T | T | T | | T | F | T | T | F | F | F | T | T | F | F | F | F | F | | T | F | F | T | T | F | F | F | F | F | F | F | T | F | | F | T | T | F | F | F | F | F | F | T | T | T | F | F | | F | T | F | F | T | T | F | F | F | T | T | T | F | T | | F | F | T | T | F | F | F | T | T | T | T | T | F | F | | F | F | F | T | T | F | F | F | F | T | F | F | F | F | 6. **Summary:** - a is true when either $\sim q \wedge r$ or $p \wedge (q \wedge \sim r)$ is true. - b is true except when $p \to q$ is true but $q \vee r$ is false. - 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.