If `P_(n)` is the set of first n prime numbers, then `underset(n=3)overset(10)capP_(n)` is

Options

A. {3,5,7,11,13,17,19}
B. {2,3,5}
C. {2,3,5,7,11,13,17}
D. {3,5,7}

Explanation

To solve the problem, we need to find the intersection of the sets of the first n prime numbers from n = 3 to n = 10, denoted as \( \bigcap_{n=3}^{10} P_n \).

### Step-by-Step Solution:

1. **Identify the Sets**:
- \( P_3 \): The set of the first 3 prime numbers = {2, 3, 5}
- \( P_4 \): The set of the first 4 prime numbers = {2, 3, 5, 7}
- \( P_5 \): The set of the first 5 prime numbers = {2, 3, 5, 7, 11}
- \( P_6 \): The set of the first 6 prime numbers = {2, 3, 5, 7, 11, 13}
- \( P_7 \): The set of the first 7 prime numbers = {2, 3, 5, 7, 11, 13, 17}
- \( P_8 \): The set of the first 8 prime numbers = {2, 3, 5, 7, 11, 13, 17, 19}
- \( P_9 \): The set of the first 9 prime numbers = {2, 3, 5, 7, 11, 13, 17, 19, 23}
- \( P_{10} \): The set of the first 10 prime numbers = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

2. **Find the Intersection**:
- We need to find \( P_3 \cap P_4 \cap P_5 \cap P_6 \cap P_7 \cap P_8 \cap P_9 \cap P_{10} \).
- The intersection of these sets will include only the elements that are present in all sets.

3. **List the Elements**:
- The common elements in all sets from \( P_3 \) to \( P_{10} \) are {2, 3, 5}.

4. **Conclusion**:
- Therefore, \( \bigcap_{n=3}^{10} P_n = \{2, 3, 5\} \).

### Final Answer:
The intersection \( \bigcap_{n=3}^{10} P_n \) is \( \{2, 3, 5\} \).