Introduction / Context:
This question mixes particular statements with a universal negative. The goal is to identify the conclusion(s) that necessarily follow, not just those that could be true in some imagined case.
Given Data / Assumptions:
- P ∩ S ≠ ∅ (some pearls are stones)
- S ∩ D ≠ ∅ (some stones are diamonds)
- D ∩ G = ∅ (no diamond is a gem)
Concept / Approach:
Universal negatives transfer across equivalence: 'No diamond is a gem' is the same as 'No gem is a diamond'. However, the premises do not relate pearls to gems at all, so any assertion about gems and pearls must be treated cautiously.
Step-by-Step Solution:
II: Some gems are diamonds. Directly contradicts D ∩ G = ∅; therefore II is false.III: No gem is a diamond. This is the contrapositive form of the given universal negative and does follow.I: Some gems are pearls. There is no link connecting pearls to gems, so I does not follow.IV: No gem is a pearl. Also not compelled by the premises; pearls unrelated to gems could or could not overlap.
Verification / Alternative check:
Build two consistent models: (A) Some pearls are gems (still no diamond is a gem), or (B) No pearl is a gem. Both satisfy the premises, showing neither I nor IV is necessary. Only III is guaranteed.
Why Other Options Are Wrong:
A and B: Include conclusions about gems–pearls that are not forced.C or D: Use either–or structures that presuppose necessity where there is none, and in C include II which is impossible.
Common Pitfalls:
Treating unrelated sets as if they must overlap or be disjoint; overlooking that the only guaranteed universal here is about diamonds and gems.
Final Answer:
None of these
Discussion & Comments