Statements and conclusions – cups, vegetables and pens: Statements: I. All cups are vegetables. II. All vegetables are pens. Conclusions: I. Some pens are vegetables. II. Some pens are cups.

Introduction / Context:
This syllogism problem involves three sets: cups, vegetables and pens. All cups are said to be vegetables, and all vegetables are said to be pens. You must determine whether the conclusions about “some pens being vegetables” and “some pens being cups” follow logically from these premises. This tests your ability to handle chained “all” relationships and to infer “some” statements from them.

Given Data / Assumptions:

Statement I: All cups are vegetables (Cups ⊂ Vegetables).
Statement II: All vegetables are pens (Vegetables ⊂ Pens).
Conclusions: I. Some pens are vegetables.
II. Some pens are cups.
We assume that each mentioned set (cups, vegetables, pens) is non-empty for standard reasoning purposes.

Concept / Approach:
When one set is fully contained inside another, and that inside set is non-empty, we can assert that “some” of the larger set are elements of the smaller set. The chain “All cups are vegetables” and “All vegetables are pens” implies Cups ⊂ Vegetables ⊂ Pens. From this nesting, we can derive that some pens are vegetables and some pens are cups, provided cups exist.

Step-by-Step Solution:
From Statement I, every cup is a vegetable. From Statement II, every vegetable is a pen. Combining these, cups are also vegetables, and those vegetables are pens. Hence, cups ⊂ vegetables ⊂ pens. Conclusion I: “Some pens are vegetables.” Since there is at least one vegetable (because there is at least one cup and cups are vegetables) and every vegetable is a pen, there must be at least one pen that is also a vegetable. Thus Conclusion I follows. Conclusion II: “Some pens are cups.” Since cups are non-empty and cups are contained within pens (through vegetables), at least one pen is also a cup. Thus Conclusion II also follows.

Verification / Alternative check:
Imagine a simple example: suppose there are 10 cups, all of which are carrots, and carrots are categorised as pens for the sake of the puzzle. Then these cups are both vegetables and pens. Therefore, some pens are vegetables (the carrots) and some pens are cups (those same objects). Any diagram or example respecting the conditions will show that both conclusions hold.

Why Other Options Are Wrong:
Saying that only one conclusion follows is incorrect because both are direct consequences of the nested set relations. Saying neither follows ignores the standard assumption that the sets involved are non-empty and that “some” simply means “at least one.” The option that suggests uncertainty about which conclusion follows is also wrong because the logic clearly supports both conclusions.

Common Pitfalls:
One pitfall is overcomplicating “some” and thinking it requires extra information beyond the subset relationship. Another is forgetting that if A ⊂ B and A is non-empty, then some B are A. Keeping a clear mental picture of nested sets helps avoid these mistakes.

Final Answer:
Both conclusions that “Some pens are vegetables” and “Some pens are cups” follow logically. The correct option is Both conclusions I and II follow.