Difficulty: Medium
Correct Answer: Only conclusion II follows
Explanation:
Introduction / Context:
This question tests your understanding of syllogisms with a focus on possibility based conclusions. You are given three nested set relationships among rings, bracelets, earrings, and necklaces. The conclusions talk about what is possible, not what is always true. Your task is to see which of the stated possibilities are compatible with the given statements and which are directly contradicted by them.
Given Data / Assumptions:
Concept / Approach:
Syllogism with possibilities requires checking compatibility. If a conclusion is a possibility, then we must see whether there exists at least one Venn diagram that satisfies all the given statements and also satisfies the content of that conclusion. If the conclusion contradicts any statement, even in the best arranged diagram, then it cannot be considered a valid possibility. The phrase “some rings are bracelets” forces an overlap between rings and bracelets, which must be respected in every diagram.
Step-by-Step Solution:
Step 1: Interpret the statements:
Step 2: From these, we can deduce that all bracelets are earrings and all earrings are necklaces, so all bracelets are necklaces. Furthermore, since some rings are bracelets, those rings are also earrings and necklaces. Thus some rings are earrings and some rings are necklaces.
Step 3: Check Conclusion I carefully. It essentially states that there is a possible arrangement in which no bracelet that is an earring is a ring. However, from Statement 1 we know that some rings are bracelets. Because all bracelets are earrings, those rings that are bracelets are automatically also earrings. That means in every valid arrangement, there is at least one bracelet which is also a ring.
Step 4: Conclusion I therefore contradicts the given fact that some rings are bracelets. You can never arrange the sets to satisfy both “some rings are bracelets” and “no bracelet is a ring” at the same time. Hence, Conclusion I is not a valid possibility.
Step 5: Now examine Conclusion II: There is a possibility that all rings are necklaces. From the statements, we already know that some rings are necklaces (those that are bracelets), but we are not told anything about rings that are not bracelets. They could lie inside or outside the necklace set without violating any statement, as long as some rings overlap bracelets.
Step 6: To see that Conclusion II is a valid possibility, construct a diagram where the entire set of rings lies inside the necklace set, and some portion of the rings intersects the bracelet set. Because bracelets are inside earrings and earrings are inside necklaces, this setup satisfies all the given statements and also makes every ring a necklace. Therefore, Conclusion II is indeed a possible situation.
Verification / Alternative check:
You can verify this by imagining concrete objects. Let there be three rings: R1, R2, and R3. Make R1 and R2 also bracelets. Since all bracelets are earrings and all earrings are necklaces, R1 and R2 are necklaces. Now place R3 somewhere inside the necklace set but outside the bracelet and earring sets. This arrangement preserves the fact that some rings are bracelets and ensures that all rings are necklaces. There is no conflict with any given statement, which confirms that Conclusion II is a valid possibility.
Why Other Options Are Wrong:
Option a: “Only conclusion I follows” is incorrect, since Conclusion I cannot be true in any arrangement that respects “some rings are bracelets”.
Option c: “Either conclusion I or conclusion II follows” is wrong because only Conclusion II is compatible with the data.
Option d: “Neither conclusion I nor conclusion II follows” is incorrect because we have just seen that Conclusion II is possible.
Option e: “Both conclusion I and conclusion II follow” is impossible because Conclusion I contradicts the basic premise that some rings are bracelets.
Common Pitfalls:
A frequent mistake is misreading “some rings are bracelets” as “some bracelets are rings” and then casually assuming that it might be turned off in another arrangement. In logical reasoning, once “some rings are bracelets” is given as a premise, it must hold in every valid model of the situation. Another pitfall is assuming that a statement about possibility can override a hard given fact, which it cannot. Always start from the premises and only then test possible arrangements that could satisfy the conclusions.
Final Answer:
Only conclusion II follows.
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