To solve the problem, let's break it down step-by-step:
1. Understand the Set A: The set [tex]\( A = \{ x : 0 < x < 10 \} \)[/tex] includes all integers greater than 0 and less than 10. Therefore, the set [tex]\( A \)[/tex] is:
[tex]\[
A = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}
\][/tex]
2. Identify Prime Numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number [tex]\( p \)[/tex] is only divisible by 1 and [tex]\( p \)[/tex].
3. Check Each Element in Set A:
- 1: By definition, 1 is not considered a prime number.
- 2: 2 is a prime number because its only divisors are 1 and 2.
- 3: 3 is a prime number because its only divisors are 1 and 3.
- 4: 4 is not a prime number because it is divisible by 1, 2, and 4.
- 5: 5 is a prime number because its only divisors are 1 and 5.
- 6: 6 is not a prime number because it is divisible by 1, 2, 3, and 6.
- 7: 7 is a prime number because its only divisors are 1 and 7.
- 8: 8 is not a prime number because it is divisible by 1, 2, 4, and 8.
- 9: 9 is not a prime number because it is divisible by 1, 3, and 9.
4. List the Prime Numbers: From the evaluation above, the prime numbers less than 10 in set [tex]\( A \)[/tex] are:
[tex]\[
\{ 2, 3, 5, 7 \}
\][/tex]
Thus, the subset [tex]\( B \)[/tex] of prime numbers is:
[tex]\[
B = \{ 2, 3, 5, 7 \}
\][/tex]
