Step 1: Understanding the Question:
The question requires us to find the specific truth values for the propositions \(p, q, r,\) and \(s\) that make the entire conditional statement \( (p \land q) \rightarrow (r \lor \neg s) \) false.
Step 2: Key Formula or Approach:
A conditional statement, also known as an implication, of the form \(A \rightarrow B\) is false only under one condition: when the antecedent (\(A\)) is True and the consequent (\(B\)) is False.
In this problem, \(A = (p \land q)\) and \(B = (r \lor \neg s)\).
Step 3: Detailed Explanation:
For the given statement to be False, we must satisfy two conditions simultaneously:
1. The antecedent \((p \land q)\) must be True.
2. The consequent \((r \lor \neg s)\) must be False.
Let's analyze the first condition:
For a conjunction (\(\land\)) to be True, both of its components must be True.
\[ (p \land q) = T \implies p = T \text{ and } q = T \]
Now, let's analyze the second condition:
For a disjunction (\(\lor\)) to be False, both of its components must be False.
\[ (r \lor \neg s) = F \implies r = F \text{ and } \neg s = F \]
From \(\neg s = F\), we can determine the truth value of \(s\).
If the negation of \(s\) is False, then \(s\) itself must be True.
\[ \neg s = F \implies s = T \]
Step 4: Final Answer:
By combining the results from the conditions, we get the required truth values:
\(p = T, q = T, r = F,\) and \(s = T\).
This corresponds to option (A).