Question

If 𝑝 → 𝑞 is false, can you determine the truth value of (~𝑝)˅(𝑝 ↔ 𝑞)? Explain your answer.

If 𝑝 → 𝑞 is true, can you determine the truth value of ~(𝑝 → 𝑞)˄~𝑝? Explain your answer. 

Answer 1

If $p \to q \equiv F$ then $p \equiv T,\,\,q \equiv F$ . Then $\left( { \sim p} \right) \vee \left( {p \leftrightarrow q} \right) \equiv F \vee \left( {T \leftrightarrow F} \right) \equiv F \vee F \equiv F$ . So, we can determine the truth value of (~𝑝)˅(𝑝 ↔ 𝑞).

f $p \to q \equiv T$ then $\sim \left( {p \to q} \right) \equiv F$ . But then $\sim \left( {p \to q} \right) \wedge \sim p \equiv F \wedge \sim p \equiv F$ . So, we can determine the truth value of ~(𝑝 → 𝑞)˄~𝑝.