Step-by-step Solution:

Quick Solution

To solve this, we need to analyze the given conditional statements (premises) and then check which of the four options is a necessary logical consequence of those premises.

1. Understanding the Premises

Let's write the given statements as simple "If... Then..." rules:

• Rule 1: If P is true → Q is true.
• Rule 2: If Q is true → R is true.
• Rule 3: If S is false → (Q is true AND R is true).

From rules 1 and 2, we can also form a logical chain: If P is true → Q is true → R is true.

2. Evaluating Each Option

Now we test each option to see if it must be true based on the rules.

• A. If S is false, then both Q and R is true.This statement is a direct, word-for-word repetition of Rule 3. Since Rule 3 is given as a fact, this statement must be true.
• B. If at least one of Q and R is true, then S is false.This is incorrect. For example, it's possible for P to be true, which would make Q and R true. In this case, Rule 3 is not triggered, so S could be true or false. Since S is not forced to be false, this statement does not follow.
• C. If P is true, then S is false.This is not necessarily true. If P is true, we know Q and R must be true. Knowing that "Q and R are true" is the *result* part of Rule 3 tells us nothing about the *condition* part ("S is false"). S could be true without creating any contradiction.
• D. If Q is true, then S is true.This is also not necessarily true. If Q is true, then R is true. As with the previous option, this doesn't force S to be either true or false. For example, S could be false, which would also make Q and R true (via Rule 3).

Final Answer: The only statement that is a guaranteed logical consequence of the premises is A, as it is a direct restatement of one of the given rules.