Difficulty: Medium
Correct Answer: Neither conclusion (I) nor conclusion (II) follows
Explanation:
Introduction / Context:
This question deals with partial overlaps among three sets: pens, pencils, and erasers. The statements tell you that some pens are pencils and some pens are erasers. You must then check whether this limited information is sufficient to conclude that some pencils are also erasers or that all erasers are pens. The exercise tests whether you can avoid drawing unjustified connections from partial overlaps.
Given Data / Assumptions:
Concept / Approach:
Each statement of the form “Some pens are pencils” indicates the existence of at least one object that belongs to both sets. However, these existence statements do not specify whether the same pen is both a pencil and an eraser or whether completely different pens are being referred to. Therefore, the safe way is to try to construct diagrams where the overlaps are separate. If you can find at least one arrangement that satisfies the statements but makes a conclusion false, then that conclusion does not logically follow.
Step-by-Step Solution:
Step 1: Draw a large set for pens.Step 2: Inside the pen set, mark a region where pens and pencils overlap, to represent “Some pens are pencils.” This region may be small and need not include all pencils or all pens.Step 3: Also inside the pen set, mark another region where pens and erasers overlap, to represent “Some pens are erasers.” Importantly, this region can be drawn disjoint from the pens that are pencils.Step 4: Analyse conclusion (I): “Some pencils are erasers.” This requires at least one object that is both a pencil and an eraser. However, we can draw the overlapping region between pens and pencils separate from the overlapping region between pens and erasers. Then there is no common element of pencils and erasers, even though both statements remain true. So conclusion (I) is not guaranteed.Step 5: Analyse conclusion (II): “All erasers are pens.” The statements only tell us that some pens are erasers. They say nothing about erasers that might lie outside the pen set. It is perfectly possible that some erasers are pens while others are not. Hence, the idea that all erasers are pens is an unjustified generalization.
Verification / Alternative check:
Consider a concrete example. Suppose there are three pen objects: P1, P2, and P3. Let P1 also be a pencil but not an eraser. Let P2 also be an eraser but not a pencil. Let P3 be only a pen. Further, suppose there are other erasers that are not pens at all. In this situation, “Some pens are pencils” is true because of P1, and “Some pens are erasers” is true because of P2. But there is no object that is both pencil and eraser, so conclusion (I) is false. Also, there are erasers that are not pens, so conclusion (II) is false. The statements hold, yet both conclusions fail.
Why Other Options Are Wrong:
Option A claims that only conclusion (I) follows, which is incorrect because we just built a valid case where no pencil is an eraser. Option B claims that only conclusion (II) follows, but we have shown erasers can exist outside the pen set. Option C claims that both conclusions follow, which is the strongest and most unjustified claim. Only option D, which states that neither conclusion follows, matches the logical analysis.
Common Pitfalls:
Learners often think that if one set overlaps with two other sets, those two must overlap with each other as well. This is a false intuition. A single set can intersect two other sets at different points without creating an intersection between those other sets. Another pitfall is to assume that a “Some A are B” statement eliminates the possibility of A elements outside B or B elements outside A, which it does not.
Final Answer:
Therefore, the correct decision is that neither conclusion (I) nor conclusion (II) follows from the given statements.
Discussion & Comments