If `A_(n)` is the set of first n prime numbers, then `underset(n=2)overset(10)UA_(n)=`

Options

A. `{2,3,5,7,11,13,17,19}`
B. `{2,3,5,7,11,13,17,19,23,29}`
C. `{3,5}`
D. `{2,3}`

Answer

To solve the problem, we need to find the union of the sets of the first n prime numbers from n = 2 to n = 10. Let's break it down step-by-step.

### Step 1: Define the sets A_n
The set \( A_n \) is defined as the set of the first n prime numbers. We will list these sets for \( n = 2 \) to \( n = 10 \).

### Step 2: List the sets
- \( A_1 = \{2\} \) (the first prime number)
- \( A_2 = \{2, 3\} \) (the first two prime numbers)
- \( A_3 = \{2, 3, 5\} \) (the first three prime numbers)
- \( A_4 = \{2, 3, 5, 7\} \) (the first four prime numbers)
- \( A_5 = \{2, 3, 5, 7, 11\} \) (the first five prime numbers)
- \( A_6 = \{2, 3, 5, 7, 11, 13\} \) (the first six prime numbers)
- \( A_7 = \{2, 3, 5, 7, 11, 13, 17\} \) (the first seven prime numbers)
- \( A_8 = \{2, 3, 5, 7, 11, 13, 17, 19\} \) (the first eight prime numbers)
- \( A_9 = \{2, 3, 5, 7, 11, 13, 17, 19, 23\} \) (the first nine prime numbers)
- \( A_{10} = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\} \) (the first ten prime numbers)

### Step 3: Find the union of the sets from \( A_2 \) to \( A_{10} \)
The union of the sets \( A_2, A_3, A_4, A_5, A_6, A_7, A_8, A_9, A_{10} \) will include all the elements from these sets. Since \( A_{10} \) contains all the prime numbers from \( A_2 \) to \( A_9 \), we can conclude:

\[ \bigcup_{n=2}^{10} A_n = A_{10} \]

### Step 4: Write the final result
Thus, the union of the sets from \( A_2 \) to \( A_{10} \) is:

\[ \bigcup_{n=2}^{10} A_n = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\} \]