Syllogism — Pens, pencils, and erasers (universal negative via subset): Statements: • All pens are pencils. • No pencil is an eraser. Conclusions: I. No eraser is a pen. II. No pen is an eraser.
-
AOnly conclusion I follows
-
BOnly conclusion II follows
-
CBoth conclusions I and II follow
-
DNeither conclusion I nor conclusion II follows
-
ENone of these
Answer
Correct Answer: Both conclusions I and II follow
Explanation
Introduction / Context:We combine a subset relation with a universal exclusion. The key is to see that exclusions apply to all subsets of the excluded class.
Given Data / Assumptions:
- Pen ⊆ Pencil.
- Pencil ∩ Eraser = ∅.
Concept / Approach:If nothing in Pencil is an Eraser, then no Pen (being a Pencil) can be an Eraser. The converse statement “No eraser is a pen” also follows because if an Eraser were a Pen, it would be a Pencil, contradicting the exclusion.
Step-by-Step:1) From Pen ⊆ Pencil and Pencil ∩ Eraser = ∅, infer Pen ∩ Eraser = ∅ → II follows.2) Suppose, for contradiction, an Eraser is a Pen. Then it is a Pencil, violating Pencil ∩ Eraser = ∅ → I follows.
Common Pitfalls:Believing only one direction holds; here both non-membership statements are equivalent because of the subset.
Final Answer:Both conclusions I and II follow.