Question

If $(p \land \sim r) \implies (q \lor r)$ is false and q and r are both false, then p is

• A. True
• B. False

Answer 1

Answer: (A)

True

Answer 2

For an implication $A \implies B$ to be false, $A$ must be true and $B$ must be false.

Given:

$$(p \land \sim r) \implies (q \lor r) \quad \text{is false, with} \quad q = \text{false} \quad \text{and} \quad r = \text{false}.$$

1. Since $q = \text{false}$ and $r = \text{false}$, the consequent becomes:

   $$q \lor r = \text{false}.$$

2. For the implication to be false, the antecedent must be true:

   $$p \land \sim r \quad \text{is true}.$$

   With $r = \text{false}$, we have $\sim r = \text{true}$. Therefore:

   $$p \land \text{true} = p \quad \text{must be true}.$$

Thus, $p$ must be true.