Difficulty: Medium
Correct Answer: Neither conclusion (I) nor conclusion (II) follows
Explanation:
Introduction / Context:
This question examines your understanding of set relations involving pens, pencils, and erasers. You are given one statement about partial overlap between pens and pencils and another statement that fully connects pencils to erasers. Two conclusions are proposed, and your task is to judge whether any of them must be true in every case consistent with the statements. The emphasis is on what is logically forced, not on what is merely possible.
Given Data / Assumptions:
Concept / Approach:
From the statements we know there is an overlapping region between pens and pencils, and that all pencils lie inside the larger set of erasers. So the set of pencils is fully contained in the eraser set. However, we do not know whether every pencil is also a pen or whether there are additional erasers that are not pencils. Logical conclusions require certainty: a conclusion follows only if it holds in every possible Venn diagram that satisfies the statements.
Step-by-Step Solution:
Step 1: Represent the pencil set as a subset of the eraser set, because all pencils are erasers.Step 2: Represent the pen set so that it overlaps the pencil set at least in one point, because some pens are pencils.Step 3: Check conclusion (I): “Some pencils are not pens.” This needs at least one pencil outside the pen region. But the given statements do not force any such extra pencil. All pencils might coincide with pens. For example, every pencil could be a pen, while still being an eraser, and the statements would still hold. Hence conclusion (I) is not necessary.Step 4: Check conclusion (II): “Some erasers are not pens.” This requires at least one eraser outside the pen set. However, it is possible that there are no erasers except those pencils which are also pens. In such a scenario, every eraser is also a pen, and conclusion (II) fails. Since the statements allow such a diagram, conclusion (II) is also not guaranteed.
Verification / Alternative check:
To verify, create a concrete model. Suppose there is exactly one object, and that object is a pen, a pencil, and an eraser at the same time. Then “Some pens are pencils” is true, and “All pencils are erasers” is also true. However, in this picture there is no pencil which is not a pen, and no eraser which is not a pen. Both conclusions (I) and (II) turn out false. Because we have found at least one valid case in which both conclusions fail, neither conclusion logically follows from the original statements.
Why Other Options Are Wrong:
Option A claims that only conclusion (I) follows, which is incorrect because we have displayed a valid arrangement where every pencil is also a pen. Option B claims that only conclusion (II) follows, but our example also shows a case where every eraser is a pen, so conclusion (II) is not forced. Option D asserts that both conclusions follow, which is clearly contradicted by the same counterexample. The only correct choice is the one that admits that neither conclusion is logically compelled.
Common Pitfalls:
Learners often confuse what is possible with what is necessary. It is easy to imagine real world situations where there are erasers unrelated to pens, or pencils that are not pens. But logical questions are stricter: unless the statement guarantees such an element in every case, we cannot accept the conclusion. Another frequent mistake is to assume that when two sets overlap, there must also be elements in each set that do not belong to the other, which is not mandatory.
Final Answer:
Therefore, based on strict syllogistic reasoning, neither conclusion (I) nor conclusion (II) follows from the given statements.
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