Logical reasoning — Pens, colors, and writing (choose the statement that must be true) Facts: Some pens don't write. All blue pens write. Some writing utensils are pens. Which statement must also be a fact?

Introduction / Context:This question requires strict adherence to the given facts without importing everyday knowledge (e.g., “pens are writing utensils”). Although that is usually true, statement 3 only says “Some writing utensils are pens,” which is weaker than “All pens are writing utensils.” We must avoid assuming more than is stated.

Given Data / Assumptions:

• Exist at least some nonfunctioning pens.
• All blue pens are functioning (they write).
• There exists at least one pen that is a writing utensil; however, it is not stated that all pens are writing utensils.

Concept / Approach:We test each candidate for necessity across all models consistent with the premises. To affirm I or II, we would need guaranteed overlap between the nonwriting pens or blue pens and the subset “writing utensils.” The facts do not force that overlap.

Step-by-Step Solution:

Verification / Alternative check:Create a model where only black pens exist, some don't write, and the subset of pens that are “writing utensils” are all functioning. All premises hold, but I and II fail; hence none is necessary.

Why Other Options Are Wrong:

• I only / I and II only / II and III only / I and III only: Each includes at least one statement not forced by the premises.

Common Pitfalls:Assuming “all pens are writing utensils” or assuming the existence of blue pens not stated by the facts.

Final Answer:None of the statements is a known fact.