Certainly! Let's go through the process of determining the sets [tex]\(P\)[/tex], [tex]\(Q\)[/tex], and [tex]\(R\)[/tex] given the set [tex]\(A\)[/tex].
1. Set [tex]\(A\)[/tex]
[tex]\[
A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}
\][/tex]
2. Set [tex]\(P\)[/tex]: elements of [tex]\(A\)[/tex] that are odd numbers
A number is odd if it is not divisible by 2. We check each element of [tex]\(A\)[/tex] to see if it is odd:
- 1 is odd
- 2 is not odd
- 3 is odd
- 4 is not odd
- 5 is odd
- 6 is not odd
- 7 is odd
- 8 is not odd
- 9 is odd
Thus, the odd numbers in [tex]\(A\)[/tex] are:
[tex]\[
P = \{1, 3, 5, 7, 9\}
\][/tex]
3. Set [tex]\(Q\)[/tex]: elements of [tex]\(A\)[/tex] that are prime numbers
A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. We check each element of [tex]\(A\)[/tex] to determine if it is prime:
- 1 is not prime
- 2 is prime
- 3 is prime
- 4 is not prime (divisible by 2)
- 5 is prime
- 6 is not prime (divisible by 2 and 3)
- 7 is prime
- 8 is not prime (divisible by 2 and 4)
- 9 is not prime (divisible by 3)
Thus, the prime numbers in [tex]\(A\)[/tex] are:
[tex]\[
Q = \{2, 3, 5, 7\}
\][/tex]
4. Set [tex]\(R\)[/tex]: elements of [tex]\(A\)[/tex] that are even numbers
A number is even if it is divisible by 2. We check each element of [tex]\(A\)[/tex] to see if it is even:
- 1 is not even
- 2 is even
- 3 is not even
- 4 is even
- 5 is not even
- 6 is even
- 7 is not even
- 8 is even
- 9 is not even
Thus, the even numbers in [tex]\(A\)[/tex] are:
[tex]\[
R = \{2, 4, 6, 8\}
\][/tex]
Hence, we have determined the elements of the sets as follows:
[tex]\[
P = \{1, 3, 5, 7, 9\}
\][/tex]
[tex]\[
Q = \{2, 3, 5, 7\}
\][/tex]
[tex]\[
R = \{2, 4, 6, 8\}
\][/tex]
