Questions: Complete the truth table. q r q ∧ r r ∧ (q ∧ r) -------------------------- T T T F O F T F F F F

Solution Steps

To complete the truth table, we need to evaluate the logical expressions for each combination of truth values for \( q \) and \( r \). The expression \( q \wedge r \) represents the logical AND operation between \( q \) and \( r \). The expression \( r \wedge (q \wedge r) \) is the logical AND operation between \( r \) and the result of \( q \wedge r \).

For each combination of truth values for \( q \) and \( r \), evaluate the logical AND operation \( q \wedge r \):

• When \( q = \text{True} \) and \( r = \text{True} \), \( q \wedge r = \text{True} \).
• When \( q = \text{True} \) and \( r = \text{False} \), \( q \wedge r = \text{False} \).
• When \( q = \text{False} \) and \( r = \text{True} \), \( q \wedge r = \text{False} \).
• When \( q = \text{False} \) and \( r = \text{False} \), \( q \wedge r = \text{False} \).

Using the results from Step 1, evaluate the expression \( r \wedge (q \wedge r) \):

• When \( q = \text{True} \) and \( r = \text{True} \), \( r \wedge (q \wedge r) = \text{True} \).
• When \( q = \text{True} \) and \( r = \text{False} \), \( r \wedge (q \wedge r) = \text{False} \).
• When \( q = \text{False} \) and \( r = \text{True} \), \( r \wedge (q \wedge r) = \text{False} \).
• When \( q = \text{False} \) and \( r = \text{False} \), \( r \wedge (q \wedge r) = \text{False} \).

Final Answer