Introduction / Context:
This is a standard syllogism question involving three categories: books, notebooks and diaries. We are given two premises and must decide which conclusions are logically forced by these premises. Such questions test your understanding of subset relationships and how to translate verbal statements like 'All X are Y' into logical diagrams or set relations.
Given Data / Assumptions:
- Statement I: All notebooks are diaries.
- Statement II: All books are notebooks.
- Conclusion I: All diaries are books.
- Conclusion II: All books are diaries.
- We assume there is at least one book and one notebook, unless otherwise contradicted, in line with typical exam conventions.
Concept / Approach:
When we read 'All books are notebooks', it means the set of books is fully contained in the set of notebooks. Similarly, 'All notebooks are diaries' means the notebooks set is contained in the diaries set. Combining these tells us how books relate to diaries. But we must be careful not to mistakenly reverse the direction of these subset relationships. The direction 'All X are Y' does not mean 'All Y are X'.
Step-by-Step Solution:
Step 1: Represent the statements as set relations. Let B be the set of all books, N be the set of all notebooks and D be the set of all diaries.
Step 2: 'All books are notebooks' translates to B ⊆ N (every book is a notebook).
Step 3: 'All notebooks are diaries' translates to N ⊆ D (every notebook is a diary).
Step 4: Combine the two subset relations. If B ⊆ N and N ⊆ D, then B ⊆ D. In words, all books are notebooks and all notebooks are diaries, so all books are diaries.
Step 5: Analyse Conclusion II: 'All books are diaries.' This is exactly the relation B ⊆ D, which we just derived. Therefore, Conclusion II definitely follows from the given statements.
Step 6: Analyse Conclusion I: 'All diaries are books.' This would mean D ⊆ B. But we only know that B ⊆ N and N ⊆ D, which together give B ⊆ D. There is no information that every diary must also be a notebook or a book. There could be diaries that are not notebooks or books at all, for example, personal diaries that are not notebooks.
Step 7: Because we can imagine diaries that are not books, while still keeping the original statements true, Conclusion I does not logically follow.
Verification / Alternative check:
Example: Suppose there are 10 diaries, out of which 4 are notebooks, and all 4 notebooks are also books. Then, 'All notebooks are diaries' and 'All books are notebooks' are both true. Still, there are 6 diaries which are not books. So 'All diaries are books' is false, but 'All books are diaries' is true.
Why Other Options Are Wrong:
Option A (Only conclusion I follows) is wrong because Conclusion I does not necessarily hold; we have no evidence that every diary is a book.
Option C (Both conclusions follow) is wrong because Conclusion I fails in many possible situations.
Option D (Neither conclusion follows) is wrong because we clearly established that all books are diaries (Conclusion II) based on transitivity of subsets.
Common Pitfalls:
A very common error is to treat 'All X are Y' as if it were 'All Y are X', which is logically incorrect. This is reversing the subset relation.
Another pitfall is ignoring transitivity: if each book is a notebook and each notebook is a diary, then each book automatically becomes a diary.
Final Answer:
Only the statement that all books are diaries is forced by the given information. Hence, Only conclusion II follows.
Discussion & Comments