Which of the following is a free-living nitrogen fixing cyanobacteria?

Options

A. (a) Cylindrospermum
B. (b) Nostoc
C. (c) Rhodospirillum
D. (d) both a and b

Explanation

### Step-by-Step Solution:

1. **Understanding Cyanobacteria**:
- Cyanobacteria are a group of photosynthetic bacteria that can perform photosynthesis and some of them have the ability to fix nitrogen.

2. **Identifying Nitrogen-Fixing Cyanobacteria**:
- Nitrogen-fixing cyanobacteria are capable of converting atmospheric nitrogen (N2) into ammonia (NH3), which can be utilized by plants.

3. **Types of Cyanobacteria**:
- Some cyanobacteria live freely in the environment, while others form symbiotic relationships with plants.

4. **Analyzing the Options**:
- **Option A: Cylindrospermum** - This is a free-living nitrogen-fixing cyanobacterium.
- **Option B: Nostoc** - This is also a free-living nitrogen-fixing cyanobacterium.
- **Option C: Rhodospirulum** - This is not a cyanobacterium; it is a purple non-sulfur bacterium and does not fit the criteria.
- **Option D: Both A and B** - Since both Cylindrospermum and Nostoc are free-living nitrogen-fixing cyanobacteria, this option is valid.

5. **Conclusion**:
- Since both options A and B are correct, the answer to the question is **Option D: Both A and B**.