In the following questions , there are statements followed by conclusions . Choose the conclusions which must logically follow from the given statements .
Statements :
1. Some men are women
2. All women are techers
3. Some teachers are doctor .
Conclusions :
I. Some doctors are women
II. Some teachers are women
III. Some teachers are men
IV. Some doctors are men

Options

A. Only I and II
B. Only I and IV
C. Only II and III
D. Only III and IV

Answer

To solve the problem, we need to analyze the given statements and draw conclusions based on logical reasoning. Let's break it down step by step:

### Step 1: Analyze the Statements
1. **Some men are women.**
- This means there is an overlap between the sets of men and women.

2. **All women are teachers.**
- This indicates that the entire set of women falls within the set of teachers.

3. **Some teachers are doctors.**
- This suggests there is an overlap between the sets of teachers and doctors.

### Step 2: Draw Venn Diagrams
- We can represent these relationships using Venn diagrams:
- Draw a circle for men and a circle for women, indicating that some men are included in the women circle.
- Draw another circle for teachers that completely encompasses the women circle, indicating all women are teachers.
- Finally, draw a circle for doctors that overlaps with the teachers circle, indicating that some teachers are also doctors.

### Step 3: Evaluate the Conclusions
Now, we will evaluate each conclusion based on the Venn diagrams:

1. **Conclusion I: Some doctors are women.**
- This conclusion cannot be definitively drawn from the statements. While some teachers are doctors, and all women are teachers, we cannot conclude that some doctors are women because the overlap is not guaranteed.

2. **Conclusion II: Some teachers are women.**
- This conclusion is true. Since all women are teachers, it follows that some teachers (specifically, those who are women) exist.

3. **Conclusion III: Some teachers are men.**
- This conclusion is also true. Since some men are women and all women are teachers, there must be some teachers who are men as well.

4. **Conclusion IV: Some doctors are men.**
- Similar to Conclusion I, we cannot definitively conclude this from the statements. While some teachers are doctors, we cannot ascertain that some of those doctors are men.

### Step 4: Final Answer
Based on the analysis:
- **Conclusions II and III are valid.**
- Therefore, the answer is that the conclusions that must logically follow are **II and III only**.

### Summary of Conclusions
- **Conclusion II:** True (Some teachers are women)
- **Conclusion III:** True (Some teachers are men)
- **Conclusion I:** False (Some doctors are women)
- **Conclusion IV:** False (Some doctors are men)