Logical reasoning — Set relations (choose the statement that must be true) Facts: All drink mixes are beverages. All beverages are drinkable. Some beverages are red. Which statements must also be facts? I: Some drink mixes are red. II: All beverages are drink mixes. III: All red drink mixes are drinkable.

Introduction / Context:
This problem involves categorical logic with set inclusions. We are given subset and universal statements and must infer what is necessarily true without assuming anything beyond the facts. The phrasing “must be a fact” requires truth in every model consistent with the premises.

Given Data / Assumptions:

• D ⊆ B (all drink mixes are beverages).
• B ⊆ K (all beverages are drinkable).
• Some B are red.

Concept / Approach:
Use transitivity of subset relations and beware of illicit conversion (reversing “all” statements). Also, “some B are red” does not imply anything about “some D are red,” unless we know a red element specifically lies in D.

Step-by-Step Solution:

Verification / Alternative check:
Chain reasoning: red drink mix → drink mix → beverage → drinkable. The color does not break the inclusion chain; hence “red drink mix” is still drinkable.

Why Other Options Are Wrong:

• I and II only / II only / I and III only: These rely on I or II, each of which is not compelled by the premises.
• None of the statements is a known fact.: Incorrect because III is certain.

Common Pitfalls:
Illicit conversion of “all” statements and assuming a “some” statement distributes into a subset without explicit evidence.

Final Answer:
III only