Difficulty: Medium
Correct Answer: Only conclusion II follows
Explanation:
Introduction / Context:
This is a syllogism problem involving four sets: pages, papers, books, and pencils. You are given three categorical statements and asked to decide which of the two possibility based conclusions follows. The main challenge is to understand what is rigidly fixed by the statements and what is still flexible, allowing different valid Venn diagrams. A conclusion that talks about possibility is valid if there exists at least one arrangement satisfying all the statements in which that conclusion also holds.
Given Data / Assumptions:
Concept / Approach:
To handle such questions, draw or imagine Venn diagrams representing the sets and their relationships. “Some pages are papers” enforces an overlap between pages and papers. “No page is a book” forces the page set to be completely outside the book set. “All books are pencils” makes the book set a subset of the pencil set. A possibility conclusion is valid if you can draw at least one diagram where it holds without breaking any given statement.
Step-by-Step Solution:
Step 1: Interpret the statements:
Step 2: Notice that nothing in the statements explicitly links books to papers or pencils to papers. The only restriction is that pages and books are disjoint, and some pages lie inside papers.
Step 3: Examine Conclusion I: “All books can never be papers.” For this to be a definite conclusion, it would have to be impossible for any arrangement to make all books papers. But the only restriction we have is that pages are not books. Books might still be papers, as long as the books lie in a region of the paper set where there are no pages.
Step 4: Construct an arrangement to test this. Place the paper set as a large region. Inside it, draw a small region for pages on one side, and another disjoint region for books on another side. Ensure that the page region does not overlap the book region to satisfy “no page is a book”. Because all books lie in papers in this diagram, all books are papers without breaking any conditions. This shows that it is not true that “all books can never be papers”. So Conclusion I does not follow.
Step 5: Now check Conclusion II: “It is possible that all pencils are papers.” From the statements, we know that all books are pencils, but we do not know anything about pencils that are not books. There is nothing preventing the pencil set from being placed completely inside the paper set, as long as the page and book constraints are respected.
Step 6: Build a diagram where all pencils are papers. Let the paper set be the largest set. Place the pencil set entirely inside the paper set. Inside the pencil set, place the book set. On a separate part of the paper set, place the pages such that they overlap with papers but do not touch books. This satisfies:
Thus Conclusion II is a valid possibility.
Verification / Alternative check:
The key is to see that the statements say nothing that forbids pencils from being entirely inside the paper set. Nor do they forbid books from being papers, as long as they are not pages. Therefore, the most flexible set is the pencil set, which you can place wholly inside the paper set without conflict. This confirms that Conclusion II, which talks about a possible arrangement, is logically correct, while Conclusion I makes a strong impossibility claim that the data do not support.
Why Other Options Are Wrong:
Option a: “Only conclusion I follows” is incorrect because Conclusion I has been shown not to be forced and, in fact, is contradicted by a valid arrangement.
Option c: “Either conclusion I or conclusion II follows” is wrong because only Conclusion II passes the possibility test.
Option d: “Neither conclusion I nor conclusion II follows” is incorrect since there clearly exists an arrangement where all pencils are papers.
Option e: “Both conclusion I and conclusion II follow” is impossible because Conclusion I is not supported by the premises.
Common Pitfalls:
Learners often confuse “no page is a book” with “no book is a paper”, which is not stated. Another frequent error is assuming that a lack of direct connection between two sets means they can never overlap, when actually it means they may or may not overlap. For possibility questions, always ask “can I draw at least one picture where this holds and all statements are still true” rather than “is it always true”.
Final Answer:
Only conclusion II follows.
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