To solve the problem of finding the complement of the set [tex]\( A \)[/tex] with respect to the universal set [tex]\( U \)[/tex], we need to follow these steps:
1. Identify the Universal Set [tex]\( U \)[/tex]:
[tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)[/tex]
2. Identify the Set [tex]\( A \)[/tex]:
[tex]\( A = \{1, 3, 5, 7, 9\} \)[/tex]
3. Understand the Definition of the Complement:
The complement of [tex]\( A \)[/tex], denoted as [tex]\( A' \)[/tex], includes all the elements that are in the universal set [tex]\( U \)[/tex] but not in the set [tex]\( A \)[/tex].
4. Determine the Elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]:
- The universal set [tex]\( U \)[/tex] contains the elements \{1, 2, 3, 4, 5, 6, 7, 8, 9\}.
- The set [tex]\( A \)[/tex] contains the elements \{1, 3, 5, 7, 9\}.
Now, we need to find which elements in [tex]\( U \)[/tex] do not appear in [tex]\( A \)[/tex]:
- Check element by element:
- [tex]\( 1 \)[/tex] is in [tex]\( A \)[/tex]
- [tex]\( 2 \)[/tex] is not in [tex]\( A \)[/tex]
- [tex]\( 3 \)[/tex] is in [tex]\( A \)[/tex]
- [tex]\( 4 \)[/tex] is not in [tex]\( A \)[/tex]
- [tex]\( 5 \)[/tex] is in [tex]\( A \)[/tex]
- [tex]\( 6 \)[/tex] is not in [tex]\( A \)[/tex]
- [tex]\( 7 \)[/tex] is in [tex]\( A \)[/tex]
- [tex]\( 8 \)[/tex] is not in [tex]\( A \)[/tex]
- [tex]\( 9 \)[/tex] is in [tex]\( A \)[/tex]
5. List the Elements Not in [tex]\( A \)[/tex]:
From the above step, the elements that are in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex] are:
\{2, 4, 6, 8\}
6. State the Complement of [tex]\( A \)[/tex]:
The complement of the set [tex]\( A \)[/tex], denoted as [tex]\( A' \)[/tex], is:
[tex]\[
A' = \{2, 4, 6, 8\}
\][/tex]
Therefore, the set [tex]\( A' \)[/tex] is [tex]\(\{2, 4, 6, 8\}\)[/tex].
