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Bartlett Patel
02/24/2024 · Elementary School

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

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To determine which logical expression has a truth value of false when \( p \) and \( q \) are true (T) and \( r \) is false (F), we can evaluate common logical operations involving these propositions. 1. **Conjunction (AND)**: \( p \land q \) - Truth value: \( T \land T = T \) 2. **Disjunction (OR)**: \( p \lor r \) - Truth value: \( T \lor F = T \) 3. **Negation (NOT)**: \( \neg r \) - Truth value: \( \neg F = T \) 4. **Implication (IF...THEN)**: \( p \to r \) - Truth value: \( T \to F = F \) 5. **Biconditional (IF AND ONLY IF)**: \( p \leftrightarrow r \) - Truth value: \( T \leftrightarrow F = F \) From the evaluations above, the expressions \( p \to r \) and \( p \leftrightarrow r \) both have a truth value of false. Thus, the logical expressions that have a truth value of false when \( p \) and \( q \) are true and \( r \) is false are: - \( p \to r \) - \( p \leftrightarrow r \) If you have specific expressions in mind, please provide them for a more targeted evaluation.

Quick Answer

The expressions \( p \to r \) and \( p \leftrightarrow r \) have a truth value of false when \( p \) and \( q \) are true and \( r \) is false.
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