Syllogism – Two premises about subsets of the same superset: Statements: (a) Some politicians are social workers. (b) All doctors are social workers. Conclusions: I) Some doctors are politicians. II) Some social workers are doctors as well as politicians. Identify the necessarily true option.

Introduction / Context:
This question tests whether you can resist inferring an overlap between two distinct subsets (Doctors and Politicians) that both lie inside the same superset (Social Workers). Without an explicit bridge, their intersection can be empty.

Given Data / Assumptions:

• Some Politicians are Social Workers (∃ Pol ∩ SW).
• All Doctors are Social Workers (Doc ⊆ SW).
• No premise asserts that any Doctor is a Politician or that the two subsets meet.

Concept / Approach:
Conclusion I requires Doc ∩ Pol ≠ ∅, which is not guaranteed: two subsets of the same superset can be disjoint. Conclusion II is even stronger; it requires existence of an element that is simultaneously Doctor and Politician (and thus a Social Worker), which fails for the same reason.

Step-by-Step Solution:
Build a countermodel: Let SW contain two disjoint regions—one holds all Doctors, the other contains the specific Politicians that are Social Workers. Then Doc ∩ Pol = ∅ while both premises hold.Therefore I does not necessarily follow. Because II requires exactly the same intersection (and then restates that such elements are SW), II also does not follow.

Verification / Alternative check:
Consider cardinalities: let Doc = {d1, d2}, Pol ∩ SW = {p1}, and ensure these sets are disjoint within SW. Both premises are satisfied; neither conclusion holds. This demonstrates non-necessity rigorously.

Why Other Options Are Wrong:

• Only I / Only II / Both: each wrongly assumes Doc ∩ Pol must be non-empty.
• None of these: not appropriate because one of the explicit options ('Neither...') is correct.

Common Pitfalls:
Assuming that sharing a superset forces overlap. In syllogisms, intersections must be stated or deduced; they are not automatic.

Final Answer:
Neither Conclusion I nor II follows.