Given that no purse is made of cloth and all purses are made of leather, which conclusions about leather, cloth and purses must follow?

Introduction / Context:
This question again involves categorical statements (syllogisms) about three sets: purses, leather items and cloth items. We are told that no purse is cloth and that all purses are leather. We must determine which conclusions about leather, cloth and purses are logically forced by these statements. This tests your ability to reason about set inclusion and exclusion without mixing up directions or making invalid generalisations.

Given Data / Assumptions:

• Statement 1: No purse is cloth.
• Statement 2: All purses are leather.
• Conclusion I: No leather is cloth.
• Conclusion II: Some leather items are cloth.
• Conclusion III: Some leather items are purses.

Concept / Approach:
Let us represent the sets:

• P = set of all purses
• L = set of all leather items
• C = set of all cloth items

'All purses are leather' means P ⊆ L. 'No purse is cloth' means P ∩ C = ∅. We must check each conclusion in turn and see whether it must hold in every possible arrangement of these sets that respects the given statements.

Step-by-Step Solution:
Step 1: From 'All purses are leather' (P ⊆ L), every purse belongs to the leather set. However, this statement does not say anything about items in L that are not purses. Step 2: From 'No purse is cloth' (P ∩ C = ∅), we know that there is no item which is both a purse and cloth. But again, this does not tell us about other leather items or other cloth items outside the purse category. Step 3: Check Conclusion I: 'No leather is cloth.' This claims that the entire leather set L and cloth set C do not overlap (L ∩ C = ∅). The given information only tells us that P, which is a subset of L, does not overlap with C. It is still possible that some other leather items (not purses) could be made of cloth in some hypothetical scenario, depending on how we interpret 'leather' and 'cloth' in the abstract. Logically, nothing stops L ∩ C from containing items that are not purses. Therefore, Conclusion I is not forced by the statements. Step 4: Check Conclusion II: 'Some leather items are cloth.' This claims that L and C do overlap (L ∩ C is non-empty). But as we just saw, the given information is compatible with both possibilities: they might overlap or they might not. So we cannot assert this as a must. Conclusion II also does not logically follow. Step 5: Check Conclusion III: 'Some leather items are purses.' Since all purses are leather, if we assume that purses exist at all, then at least some leather items are purses. Exam questions of this type usually assume that the subject set (here, purses) is non-empty, otherwise the statements would be trivial. So, if P is non-empty and P ⊆ L, then there is at least one element in both P and L, meaning some leather items are purses. Step 6: Therefore, Conclusion III follows: there exist leather items that are purses.
Verification / Alternative check:
Example where Conclusions I and II fail but the statements hold: Let there be leather jackets that are partly cloth-lined, counted in both L and C. Also let some purses be pure leather and not cloth. Then 'All purses are leather' and 'No purse is cloth' remain true, but some leather items are cloth, so Conclusion I is false and Conclusion II is true. This shows that nothing in the statements forbids such a situation, hence Conclusions I and II do not necessarily follow. However, as long as at least one purse exists and all purses are leather, then some leather items are purses. Conclusion III must hold under the usual exam assumption that the set of purses is non-empty.
Why Other Options Are Wrong:
Option A (Only conclusion I follows) is wrong because we cannot rule out the possibility that some leather items are also cloth, as long as they are not purses. Option B (Only conclusion II follows) is wrong because the existence of leather items that are cloth is not guaranteed by the statements. Option D (All the above) is wrong because Conclusions I and II are not logically forced.
Common Pitfalls:
A very common mistake is to jump from 'No purse is cloth' and 'All purses are leather' to 'No leather is cloth', which overgeneralises the restriction placed only on purses. Another frequent error is to ignore the usual assumption of existence (that there is at least one purse), which is needed to make sense of 'All purses are leather' in the context of typical exam questions.
Final Answer:
The only statement that must be true in all valid scenarios is that some leather items are purses. Therefore, only conclusion III follows.