Question

Three statements are given below followed by their conclusions. Choose the best option that fits :
Statements:
1. All lemons are plums.
2. All plums are dates.
3. Some dates are mangoes.

Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.

• A. Only conclusions II and III follow
• B. Only conclusions I and IV follow
• C. Only conclusions I and II follow
• D. Both conclusions III and IV follow

Hint:

In syllogisms, when sets are nested:
- "All A are B" and "All B are C" always leads to the definite conclusion "All A are C".
- The statement "Some A are B" is symmetric and always implies "Some B are A".

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This question is based on syllogisms. We need to evaluate the validity of four conclusions based on three given statements.

Step 2: Key Formula or Approach:
We can represent the relationships between different sets (lemons, plums, dates, and mangoes) using a Venn diagram:
- "All A are B" implies that the circle representing set A lies completely inside the circle representing set B.
- "Some A are B" implies that the circles representing sets A and B overlap with each other.

Step 3: Detailed Explanation:
1. Let the sets of lemons, plums, dates, and mangoes be represented by \( L \), \( P \), \( D \), and \( M \) respectively.
2. Statement 1 states: "All lemons are plums." This means that the set \( L \) is completely inside the set \( P \) (\( L \subseteq P \)).
3. Statement 2 states: "All plums are dates." This means that the set \( P \) is completely inside the set \( D \) (\( P \subseteq D \)).
4. Combining these two nested statements, since \( L \subseteq P \) and \( P \subseteq D \), it is mathematically certain that \( L \subseteq D \). This means "All lemons are dates." Therefore, Conclusion III is valid.
5. Statement 3 states: "Some dates are mangoes." This means there is an intersection between the set of dates \( D \) and the set of mangoes \( M \) (\( D \cap M \neq \emptyset \)).
6. This intersecting set of mangoes does not necessarily overlap with the subset of plums \( P \) or the subset of lemons \( L \).
7. Let us analyze Conclusion I: "Some lemons are mangoes." Since there is no definite overlap between \( L \) and \( M \), this is not necessarily true. Therefore, Conclusion I does not follow.
8. Let us analyze Conclusion II: "Some mangoes are plums." Since there is no definite overlap between \( M \) and \( P \), this is also not necessarily true. Therefore, Conclusion II does not follow.
9. Let us analyze Conclusion IV: "Some mangoes are dates." Since "Some dates are mangoes" implies a mutual intersection, it is always true that "Some mangoes are dates." Therefore, Conclusion IV is valid.
10. Thus, only Conclusions III and IV follow.

Step 4: Final Answer:
Both conclusions III and IV follow, which corresponds to option (D).