Difficulty: Easy
Correct Answer: Only conclusion I follows
Explanation:
Introduction / Context:
This is a straightforward syllogism problem about clocks, watches, and alarms. The statements describe subset relationships between clocks and watches and a partial overlap between clocks and alarms. Your goal is to determine whether each proposed conclusion necessarily follows from these relations when the statements are treated as completely true.
Given Data / Assumptions:
Concept / Approach:
When we say “All clocks are watches,” we mean that the set of clocks is entirely contained within the set of watches. The second statement, “Some clocks are alarm,” means there exists at least one object that is both a clock and an alarm. By combining these facts, we can identify at least one object that is at the same time a clock, a watch, and an alarm. This clearly supports a “some” type conclusion. However, we must be careful not to overgeneralize this to “all watches.”
Step-by-Step Solution:
Step 1: Draw a large set for watches.Step 2: Place the clocks set entirely inside the watches set because all clocks are watches.Step 3: Represent alarms as another set that intersects the clocks set in at least one region because some clocks are alarm.Step 4: Consider conclusion I: “Some alarms are watches.” The clocks that are alarms are also watches because all clocks are watches. Therefore, at least one object that is an alarm is also a watch. So conclusion I definitely follows.Step 5: Consider conclusion II: “All watches are alarm.” This would require the entire set of watches to be inside the set of alarms. However, the statements only guarantee that some clocks (a subset of watches) are alarms. Other watches, which are not clocks, are unconstrained relative to alarms and can be non alarm watches. Hence conclusion II is not forced by the statements.
Verification / Alternative check:
We can verify the logic with a numerical example. Suppose there are 10 watches. Among them, 4 are clocks and 6 are wristwatches without any clock label. Out of the 4 clocks, 2 are alarms. So “All clocks are watches” holds, since all 4 clocks are among the 10 watches. “Some clocks are alarm” also holds, because 2 clocks are alarms. However, not all 10 watches are alarms; the 6 wristwatches are not alarms. In this example, conclusion I is true because those 2 alarm clocks are both alarms and watches. Conclusion II is false because many watches are not alarms. Since the statements are satisfied while conclusion II fails, conclusion II cannot be accepted as a necessary logical outcome.
Why Other Options Are Wrong:
Option B claims that only conclusion II follows, but we have just constructed a scenario where conclusion II is false. Option C says both conclusions follow, which again is contradicted by the counterexample. Option D claims that neither conclusion follows, ignoring the clear reasoning that some alarms are necessarily watches. Only option A, which accepts conclusion I and rejects conclusion II, is compatible with proper logical reasoning.
Common Pitfalls:
Students sometimes misinterpret “All clocks are watches” as “All watches are clocks,” reversing the subset relationship. This leads to the mistaken belief that if some clocks are alarms, then all watches must be alarms. Always remember that “All A are B” does not mean “All B are A.” Another pitfall is neglecting the existence information in “some” statements, which is crucial to support conclusion I.
Final Answer:
Thus, the logically correct result is that only conclusion I follows from the given statements.
Discussion & Comments