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Statements: All pens are pencils. All pencils are erasers. Conclusions: I. All pens are erasers. II. Some erasers are pens.
Only I follows
Only II follows
Both I and II follow
Neither I nor II follows
Both I and II follow
Think of the statements as nested circles: Pens are inside Pencils, and Pencils are inside Erasers; therefore, anything inside Pens is automatically inside Erasers, and any intersection with Pens is also an intersection with Erasers.
Think of the statements as nested circles: Pens are inside Pencils, and Pencils are inside Erasers; therefore, anything inside Pens is automatically inside Erasers, and any intersection with Pens is also an intersection with Erasers.
Visualization
Draw three circles where the circle for 'Pens' is entirely inside the circle for 'Pencils', and the circle for 'Pencils' is entirely inside the circle for 'Erasers'. This creates a nested structure: Pens < Pencils < Erasers.
Evaluating Conclusion I
Since the set of Pens is a subset of Pencils, and Pencils is a subset of Erasers, by the transitive property, the set of Pens is a subset of Erasers. Thus, All pens are erasers is TRUE.
Evaluating Conclusion II
Since all pens are erasers, it implies that the region occupied by pens is contained within the region of erasers. Therefore, any part of the pens set is also part of the erasers set. Thus, Some erasers are pens is TRUE.
A: Incorrect because it ignores that Conclusion II also logically follows from the Venn diagram. B: Incorrect because it ignores that Conclusion I also logically follows from the Venn diagram. D: Incorrect because both conclusions are logically supported by the given statements.
C is correct because the transitive nature of the 'All' relationship ensures that if A is inside B and B is inside C, then A is entirely within C, and consequently, a portion of C must contain A.
When you see all statements starting with 'All', remember that the conclusions will hold true for all levels of nesting, but be careful with 'Some' conversions where the reverse might not always hold unless explicitly linked.