Introduction / Context:
This problem combines two universal statements with one particular statement. We must be precise about what the particular fact ('some bricks are ropes') guarantees and what it does not.
Given Data / Assumptions:
- R ⊆ B (all rods are bricks)
- ∃ element in B ∩ Ro (some bricks are ropes)
- Ro ⊆ D (all ropes are doors)
Concept / Approach:
Transitivity helps only when sets link directly. The fact that some bricks are ropes does not mean any rods are ropes, since rods are only a subset of bricks and the 'rope' subset could be disjoint from the 'rod' subset inside bricks.
Step-by-Step Solution:
II: Some doors are bricks. True. The bricks that are ropes (given) are doors, since all ropes are doors. Hence there exist doors that are bricks.I: Some rods are doors. Not necessary. It would require that the rod subset of bricks intersects the rope subset. The premises do not force this overlap.III: Some rods are not doors. Also not necessary. It is possible that all rods sit inside the rope subset, making all rods doors. The premises do not fix this either way.IV: All doors are ropes. False. We only know all ropes are doors, not the converse.
Verification / Alternative check:
Model A: Place rods inside bricks away from the rope subset. Then I fails and III holds. Model B: Place rods wholly inside the rope subset. Then I holds and III fails. In both models II is true and IV is false. Since none of the provided options list only II, the correct choice is 'None of these'.
Why Other Options Are Wrong:
A and B: Include I and/or III as necessary, which they are not.C and D: Invoke an either–or along with II or IV, but IV is false and the either–or between I and III is not compelled.
Common Pitfalls:
Assuming a subset of a subset must overlap another subset; confusing 'all X are Y' with 'all Y are X'.
Final Answer:
None of these
Discussion & Comments