Difficulty: Easy
Correct Answer: Only conclusion I follows
Explanation:
Introduction / Context:
This is a straightforward syllogism problem dealing with three sets: books, notebooks, and diaries. You must decide which of the conclusions follow from the given statements when they are treated as absolutely true. The key is to interpret the set relations accurately and see what they imply for possible overlaps between notebooks and diaries.
Given Data / Assumptions:
Concept / Approach:
We use Venn diagram logic and subset rules. If one set is a subset of another, and that larger set is disjoint from a third set, then the subset must also be disjoint from that third set. Conversely, if a subset is proposed to lie inside a set that is known to be disjoint from its super set, the proposal contradicts the premises. We must test both conclusions against these rules.
Step-by-Step Solution:
Step 1: From “No books are notebooks,” we know that Books ∩ Notebooks = empty. There is no element that is both a book and a notebook.
Step 2: From “All diaries are books,” we know that Diaries is a subset of Books. Every diary is definitely a book.
Step 3: Consider Conclusion I: “No notebooks are diaries.” If all diaries are books, and no books are notebooks, then no diary can be a notebook, because that would make a diary both a book and a notebook. Formally, since Diaries ⊂ Books and Books ∩ Notebooks = empty, it follows that Diaries ∩ Notebooks = empty. So Conclusion I is logically correct and must follow.
Step 4: Now evaluate Conclusion II: “All diaries are notebooks.” This claims that the set Diaries is a subset of Notebooks. However, we already know that Diaries are a subset of Books, and Books and Notebooks do not intersect at all. If any diary were a notebook, that diary would be forced to lie in both Books and Notebooks, which is impossible.
Step 5: Because Conclusion II directly contradicts the given relationship between Books and Notebooks, it cannot follow. No diagram can satisfy both premises and the claim that all diaries are notebooks. Therefore, Conclusion II is false.
Verification / Alternative check:
Draw a Venn diagram with three sets: Books, Notebooks, and Diaries. Place the Diaries circle entirely inside the Books circle. Now, because Books and Notebooks are disjoint, the Notebooks circle must lie outside the Books circle with no overlap. As a result, it is visually clear that no part of the Diaries circle can touch the Notebooks circle. This means no diary is a notebook, confirming Conclusion I. At the same time, there is no way to redraw the diagram so that Diaries lies inside Notebooks while still staying inside Books and not overlapping with Notebooks. This reinforces that Conclusion II is impossible.
Why Other Options Are Wrong:
Option b: “Only conclusion II follows” is incorrect because Conclusion II conflicts with the premises.
Option c: “Both conclusions I and II follow” is impossible, since the two conclusions contradict each other in light of the given data.
Option d: “Neither conclusion I nor conclusion II follows” is wrong because Conclusion I is clearly forced by the subset and disjointness relationships.
Option e: “Either conclusion I or conclusion II follows” wrongly suggests that exactly one of them holds in different scenarios, but in fact only Conclusion I is compatible with the premises.
Common Pitfalls:
A common error is to assume that if two sets do not overlap, a third set linked to one of them might still overlap with the other, even when the logical structure prevents it. Another mistake is not fully visualizing the subset relationships. Always remember that if a set is completely inside another set, and that larger set has no overlap with a third set, then the smaller set also cannot overlap with that third set.
Final Answer:
Only conclusion I follows.
Discussion & Comments