Statements: • All chairs are keys. • All keys are balloons. • Some balloons are mirrors. • Some mirrors are desks. Conclusions: I. Some desks are keys. II. Some balloons are mirrors. III. Some mirrors are balloons. Choose the option that must follow.

Introduction / Context:Two universal inclusions are followed by two existentials forming a direct equivalence. We check which consequences are guaranteed without assuming that the desk-mirror items are among the keys.

Given Data / Assumptions:

• Chairs ⊆ Keys ⊆ Balloons.
• ∃b_m ∈ Balloons ∩ Mirrors.
• ∃m_d ∈ Mirrors ∩ Desks.

Concept / Approach:II and III are the same overlap phrased both ways and are explicitly given by the premise “Some balloons are mirrors.” I would need that some Desk is also a Key via Mirrors, which is not required: not all Balloons are Keys, so a Mirror can be a Balloon without being a Key.

Step-by-Step Solution:• II and III: Guaranteed by the existential premise.• I: Would require a Mirror that is both a Desk and a Key. The premises do not force that identity.

Verification / Alternative check:Model the Balloons∩Mirrors element outside Keys; then II–III remain true while I fails.

Why Other Options Are Wrong:They include I, which assumes a non-forced overlap.

Common Pitfalls:Assuming mirrors that are balloons are also keys; that is not assured.

Final Answer:Only II and III follow.