Problem 1 - Entrance Test

Statements: 1. All pens are pencils. 2. Some pencils are erasers. Conclusions: I. Some pens are erasers. II. No pen is an eraser.

A. A. Only conclusion I follows.B. B. Only conclusion II follows.C. C. Either conclusion I or II follows.D. D. Neither conclusion I nor II follows.E. E. Both conclusions I and II follow.

Correct: C

Let's represent the statements using Venn diagrams: Statement 1: All pens are pencils. (Pens circle is entirely inside Pencils circle) Statement 2: Some pencils are erasers. (Erasers circle overlaps with a portion of the Pencils circle). Now, let's analyze the conclusions: I. Some pens are erasers. From the diagram, it's possible that the Erasers overlap with the Pens, but it's not certain. The overlap could be only with the Pencils that are not Pens. So, Conclusion I is not necessarily true (uncertain). II. No pen is an eraser. Similarly, it's possible that Erasers do not overlap with Pens at all, but it's not certain. So, Conclusion II is also not necessarily true (uncertain). However, conclusions I (Some pens are erasers) and II (No pen is an eraser) form a complementary pair (specifically, a contradictory pair 'I' type and 'E' type propositions). If 'Some A are B' is false, then 'No A are B' must be true, and vice-versa. Since both are individually uncertain, one of them must be true. Therefore, 'Either conclusion I or II follows'.