Step 1: Understanding the Concept:
We are dealing with propositional logic. We need to evaluate the truth values of compound statements based on the given conditions.
The logical connectives are AND ($\land$), OR ($\lor$), implies ($\rightarrow$), and iff ($\leftrightarrow$).
Step 2: Key Formula or Approach:
Use truth tables or known properties:
$p \land F \equiv F$ (Anything AND False is False).
$F \leftrightarrow r \equiv T$ if and only if $r$ is False (Both sides must have same truth value).
$F \rightarrow \text{Anything} \equiv T$ (A conditional with a false premise is always true).
Step 3: Detailed Explanation:
Given:
Truth value of $q$ is False ($q = F$).
Truth value of $(p \land q) \leftrightarrow r$ is True.
First, evaluate the truth value of $(p \land q)$:
Since $q = F$, the conjunction $p \land F$ will always be False, regardless of the truth value of $p$.
So, $(p \land q) = F$.
Now substitute this into the biconditional statement:
$F \leftrightarrow r$ is True.
A biconditional statement $A \leftrightarrow B$ is true if and only if both $A$ and $B$ have the same truth value.
Since the left side is $F$ and the whole statement is True, the right side $r$ must also be $F$.
So, $r = F$.
We don't know the truth value of $p$, it could be True or False. Let's analyze the options:
(A) $p \land q$: We already established this is $F$.
(B) $p \lor r$: Since $r = F$, this becomes $p \lor F \equiv p$. Its truth value depends on $p$, so it's not necessarily True.
(C) $p \land r$: Since $r = F$, this becomes $p \land F \equiv F$.
(D) $(p \land r) \rightarrow (p \lor r)$: From our evaluation, $(p \land r) = F$.
The statement becomes $F \rightarrow (p \lor r)$.
In logic, an implication $A \rightarrow B$ is always True if the antecedent $A$ is False (vacuous truth).
Therefore, $F \rightarrow (p \lor r)$ is True, regardless of the value of $p \lor r$.
Step 4: Final Answer:
The statement $(p \land r) \rightarrow (p \lor r)$ is True.