Statements: • All belts are rollers. • Some rollers are wheels. • All wheels are mats. • Some mats are cars. Conclusions: I. Some mats are rollers. II. Some mats are belts. III. Some cars are rollers. IV. Some rollers are belts. Choose the option that must follow.

Introduction / Context:Here a universal step (wheels ⊆ mats) converts an existential about rollers∩wheels into a guaranteed intersection with mats. Other proposed overlaps require assumptions not provided.

Given Data / Assumptions:

• Belts ⊆ Rollers.
• ∃r_w ∈ Rollers ∩ Wheels.
• Wheels ⊆ Mats.
• ∃m_c ∈ Mats ∩ Cars.

Concept / Approach:Because r_w is a Wheel and all Wheels are Mats, r_w ∈ Mats, so Mats ∩ Rollers ≠ ∅ (I true). Claims II–IV either need existence of Belts (not given) or identity between the mats that are cars and the mats that are wheels (not given).

Step-by-Step Solution:• I: r_w witnesses Rollers ∩ Mats.• II: Would require that some Belt is among those Rollers that are Wheels; existence of Belts is not stated.• III: Would need the Car-Mats to be the same as the Wheel-Mats; not forced.• IV: “Some rollers are belts” needs existence of Belts; “All Belts are Rollers” does not assert any Belt exists.

Verification / Alternative check:Let Rollers contain two elements: one that is a Wheel (hence in Mats) and one that is a Belt (not a Wheel). Let Cars occupy a different part of Mats. Then only I is guaranteed.

Why Other Options Are Wrong:They infer non-forced overlaps or assume existential import for “All Belts are Rollers.”

Common Pitfalls:Confusing subset statements with existence claims.

Final Answer:Only I follows.