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

Denote: $true = 1, false = 0$.

1) If $p \to q = 0 \Longrightarrow p = 1$, $q = 0$.

Then $(\thicksim p) = 0, p \leftrightarrow q = 0, (\thicksim p) \lor (p \leftrightarrow q) = 0 \lor 0 = 0.$

2) If $p \to q = 1 \Longrightarrow p = 1, q =1$ or $p = 0, q = 0$ or $p=0, q =1.$

Then $\thicksim (p \to q) = 0, \thicksim p = 1$ or $\thicksim p = 0$.

If $\thicksim p = 1 : \thicksim (p \to q) \land \thicksim p = 0 \land 1 = 0;$

If $\thicksim p = 0 : \thicksim (p \to q) \land \thicksim p = 0 \land 0 = 0 \Longrightarrow$ No matter $\thicksim p = 0$ or $\thicksim p = 1$, the value of $\thicksim (p \to q) \land \thicksim p$ is always 0.