When \( p \) and \( q \) are both true propositions while \( r \) is false, which of the following has truth value of false? \( p \vee q \vee r \) \( p \vee q \vee \neg r \) \( (p \Rightarrow q) \wedge \neg r \) \( (\neg r \Rightarrow \neg(p \wedge q) \)

To determine which of the given propositions has a truth value of false when \( p \) and \( q \) are true and \( r \) is false, we will evaluate each expression step by step. 1. **Expression: \( p \vee q \vee r \)** - Since \( p \) is true and \( q \) is true, the disjunction \( p \vee q \) is true. - Therefore, \( p \vee q \vee r \) is true (because true OR anything is true). 2. **Expression: \( p \vee q \vee \neg r \)** - Since \( r \) is false, \( \neg r \) is true. - Thus, \( p \vee q \vee \neg r \) is true (because true OR anything is true). 3. **Expression: \( (p \Rightarrow q) \wedge \neg r \)** - The implication \( p \Rightarrow q \) is true because both \( p \) and \( q \) are true. - Since \( r \) is false, \( \neg r \) is true. - Therefore, \( (p \Rightarrow q) \wedge \neg r \) is true (true AND true is true). 4. **Expression: \( \neg r \Rightarrow \neg(p \wedge q) \)** - Since \( r \) is false, \( \neg r \) is true. - Now, \( p \wedge q \) is true (both \( p \) and \( q \) are true), so \( \neg(p \wedge q) \) is false. - Therefore, we have true \( \Rightarrow \) false, which is false. Now, we can summarize the truth values of each expression: - \( p \vee q \vee r \): true - \( p \vee q \vee \neg r \): true - \( (p \Rightarrow q) \wedge \neg r \): true - \( \neg r \Rightarrow \neg(p \wedge q) \): false Thus, the expression that has a truth value of false is: \[ \neg r \Rightarrow \neg(p \wedge q) \]