Introduction / Context:
This problem chains two universal statements and one existential. We must determine which conclusions are compelled by these relationships and the existence claim.
Given Data / Assumptions:
- Some Uniforms are Covers (∃ U ∩ C).
- All Covers are Papers (C ⊆ P).
- All Papers are Bags (P ⊆ B).
Concept / Approach:
- Transitivity: C ⊆ P and P ⊆ B implies C ⊆ B.
- The existing overlap U ∩ C travels “up” through the chain: those same elements are also Papers and Bags.
Step-by-Step Solution:
(I) Since all Covers are Papers and all Papers are Bags, necessarily all Covers are Bags. So (I) follows.(II) Because some Uniforms are Covers, those very elements are also Papers (by C ⊆ P) and Bags (by P ⊆ B). Hence there exist Bags that are simultaneously Covers, Papers, and Uniforms. So (II) follows.(III) “Some uniforms are not papers” contradicts the fact that at least some Uniforms are Covers and all Covers are Papers. The premises do not force any Uniform to be outside Papers. Therefore (III) does not follow.
Verification / Alternative check:
A quick Venn sketch showing U overlapping C, with C entirely inside P and P entirely inside B, confirms (I) and (II) and refutes (III).
Why Other Options Are Wrong:
“Only I” ignores the existential consequence in (II). “All three” wrongly includes (III). “None of these” does not match the compelled pair (I and II).
Common Pitfalls:
Missing that “some U are C” guarantees those particular U are also in P and B due to subset chaining.
Final Answer:
Only I and II follow
Discussion & Comments