Difficulty: Medium
Correct Answer: Only II follows
Explanation:
Introduction / Context:Here multiple set relations appear together. The safe path is to identify conclusions supported by a single chain using the same existential witness, and to reject conclusions that require intersections not guaranteed by the premises.
Given Data / Assumptions:
Concept / Approach:From “Some engineers are lawyers” and “All engineers are businessmen,” pick that same engineer as the witness; they are both a businessman and a lawyer. That confirms “Some businessmen are lawyers.” Intersections involving Teachers require existence of Teachers (not stated) and further overlap with Doctors or Businessmen (also not stated).
Step-by-Step Solution:
II: Choose y with y ∈ Engineers ∩ Lawyers. Since Engineers ⊆ Businessmen, y ∈ Businessmen as well. Thus Businessmen ∩ Lawyers ≠ ∅ ⇒ II follows.I: “Some teachers are doctors” needs Teachers ∩ Doctors ≠ ∅; we have no bridge creating that overlap.III: “Some businessmen are teachers” needs Businessmen ∩ Teachers ≠ ∅; again, no bridge.IV: “Some lawyers are teachers” requires ∃Teacher; the premises never assert that any teacher exists. Hence IV is not necessary.Verification / Alternative check:Construct a model with engineers (hence businessmen) who are lawyers, some doctors who are lawyers, and zero teachers. All premises hold and only II is forced.
Why Other Options Are Wrong:Any option including I, III, or IV assumes extra existence/overlap not ensured by the premises.
Common Pitfalls:Assuming existence for a class mentioned only in a universal statement (“All teachers are lawyers”) and over-connecting different “some” statements.
Final Answer:Only II follows.
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