Syllogism – Books, schools, and colleges (careful with shared supersets) Statements: • Some pens are books. • All schools are books. • Some colleges are schools. Conclusions to test: I. Some colleges are pens. II. Some pens are schools. III. Some colleges are books.

Introduction / Context:
This question illustrates a classic pitfall: when two classes are both contained in a larger class, we cannot infer that those two classes overlap each other. “All schools are books” and “some pens are books” do not guarantee any common element between schools and pens.

Given Data / Assumptions:

• Some pens are books (P ∩ B ≠ ∅).
• All schools are books (S ⊆ B).
• Some colleges are schools (C ∩ S ≠ ∅).

Concept / Approach:
Use subset propagation carefully. From “some colleges are schools” and “all schools are books,” we can confidently move those specific colleges into the books set. However, moving from pens to schools (or colleges to pens) cannot be done without an explicit connecting statement beyond the shared superset “books.”

Step-by-Step Solution:
III: “Some colleges are books.” Since some colleges are schools and every school is a book, those colleges are books. III follows.I: “Some colleges are pens.” No link connects colleges to pens except both relate to books. Shared supersets do not imply overlap. I does not follow.II: “Some pens are schools.” Again, pens and schools both sit within books, but that does not force any intersection. II does not follow.

Verification / Alternative check:
Create a model where the “pens inside books” subset is entirely separate from the “schools (and thus colleges) inside books” subset. All premises hold; only III is true. Because no option lists “Only III,” the correct choice is “None of these.”

Why Other Options Are Wrong:

• Any option pairing I or II asserts overlaps not supported by the premises.
• “All follow” overreaches; only III is compelled.

Common Pitfalls:
Assuming that two classes contained in the same larger class must overlap; ignoring that “some” statements identify only part of a set.

Final Answer:
None of these