Syllogism — chaining universal inclusions across three classes Statements: • All fruits are vegetables. • All pens are vegetables. • All vegetables are rains. Conclusions to evaluate: I. All fruits are rains. II. All pens are rains. III. Some rains are vegetables.

Introduction / Context:This syllogism features a clean chain of subset relations. The task is to apply transitivity and, where appropriate, infer safe existential statements from universal ones when nonemptiness is ensured by earlier premises.

Given Data / Assumptions:

• F = fruits, V = vegetables, R = rains, Pn = pens.
• F ⊆ V and Pn ⊆ V.
• V ⊆ R.
• F is nonempty (as a named real-world category), hence V is nonempty as well.

Concept / Approach:Transitivity of inclusion gives immediate universal conclusions about fruits and pens relative to rains. In addition, since V is nonempty and V ⊆ R, there exists at least one element that is both rain and vegetable, which justifies an existential conclusion.

Step-by-Step Solution:

Verification / Alternative check:Diagrammatically, put F and Pn inside V and then place V fully inside R. This immediately shows I and II. The nonemptiness of V guarantees at least one overlap element for III.

Why Other Options Are Wrong:

• Options B, C, and D exclude one of the valid conclusions.
• Option A claims none follows, which contradicts clear transitivity results.

Common Pitfalls:Forgetting to apply transitivity; hesitation about III due to universal premises—remember that the existence of fruits implies vegetables exist, enabling the existential claim.

Final Answer:All follow