If the statement “Some residents are voters” is true, consider the following…

Concept

An existential affirmative statement says that the two named sets share at least one member. Such a statement permits simple conversion: “Some A are B” implies “Some B are A.”

It does not establish universal inclusion, and it is incompatible with a universal statement that the two sets are disjoint.

Application

1. Let R be the set of residents and V the set of voters. “Some residents are voters” means that at least one person lies in R ∩ V.

2. The same shared person is both a voter and a resident, so conclusion 3, “Some voters are residents,” follows by simple conversion.

3. Conclusion 2 says R is entirely contained in V. One shared member does not prove that every resident belongs to V.

4. Conclusions 1 and 4 both say R and V are disjoint. That contradicts the given non-empty intersection.

Cross-check and contrast

• “Only 1 and 2” combines two universal claims that are not established by the existential statement.

• “Only 2” strengthens “some” to “all,” which is not licensed.

• “Only 3” preserves the same shared witness while reversing the order of the two classes.

• “Only 3 and 4” combines the shared-witness claim with a disjointness claim, so the pair is internally inconsistent.

Therefore, conclusion 3 alone follows, so the answer is “Only 3.”