In the following questions , there are statements followed by conclusions . Choose the conclusions which must logically follow from the given statements .
Statements :
1. Some grandmothers are mothers
2. Some mother are daughters
3. All the daughters are married women
Conclusions :
I. Some married women are mothers
II. Some daughters are grandmothers
III. No daughter is grandmother
IV. Some mothers are grandmothers

Options

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

Answer

To solve the problem, we will analyze the given statements and conclusions using logical reasoning and Venn diagrams.

### Step-by-Step Solution:

1. **Identify the Statements**:
- Statement 1: Some grandmothers are mothers.
- Statement 2: Some mothers are daughters.
- Statement 3: All daughters are married women.

2. **Draw Venn Diagrams**:
- **Venn Diagram for Statement 1**:
- Draw two circles, one for "Grandmothers" and one for "Mothers".
- Shade the area where the two circles overlap to represent "Some grandmothers are mothers".

- **Venn Diagram for Statement 2**:
- Draw two circles, one for "Mothers" and one for "Daughters".
- Shade the overlapping area to represent "Some mothers are daughters".

- **Venn Diagram for Statement 3**:
- Draw a circle for "Daughters" and a larger circle for "Married Women".
- Since all daughters are married women, the "Daughters" circle should be completely inside the "Married Women" circle.

3. **Analyze Conclusions**:
- **Conclusion I**: Some married women are mothers.
- From the Venn diagrams, we see that since some mothers are daughters and all daughters are married women, it follows that some mothers must also be married women. Thus, this conclusion is true.

- **Conclusion II**: Some daughters are grandmothers.
- This conclusion does not necessarily follow from the statements. While some mothers are daughters, and some grandmothers are mothers, it does not imply that some daughters are grandmothers. Thus, this conclusion is false.

- **Conclusion III**: No daughter is a grandmother.
- This conclusion is not necessarily true. Since some grandmothers are mothers, and some mothers are daughters, it is possible for a daughter to also be a grandmother. Thus, this conclusion is false.

- **Conclusion IV**: Some mothers are grandmothers.
- This conclusion follows from Statement 1, where we established that some grandmothers are mothers. Thus, this conclusion is true.

4. **Final Evaluation**:
- The conclusions that logically follow from the statements are:
- Conclusion I: True
- Conclusion II: False
- Conclusion III: False
- Conclusion IV: True

### Answer:
The conclusions that must logically follow from the given statements are:
- I. Some married women are mothers (True)
- IV. Some mothers are grandmothers (True)