To find the complement of the set [tex]\( A \)[/tex] within the universal set [tex]\( U \)[/tex], we follow these steps:
1. Identify the elements in the universal set [tex]\( U \)[/tex]: [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex].
2. Identify the elements in the set [tex]\( A \)[/tex]: [tex]\( A = \{2, 4, 6, 8\} \)[/tex].
3. Determine the complement of [tex]\( A \)[/tex]:
The complement of [tex]\( A \)[/tex] in [tex]\( U \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
4. List out the elements of [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]:
- [tex]\( 1 \)[/tex] is in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
- [tex]\( 3 \)[/tex] is in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
- [tex]\( 5 \)[/tex] is in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
- [tex]\( 7 \)[/tex] is in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
- [tex]\( 9 \)[/tex] is in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
- [tex]\( 10 \)[/tex] is in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
5. Write down the complement of [tex]\( A \)[/tex]:
The complement of [tex]\( A \)[/tex], denoted as [tex]\( A' \)[/tex] or [tex]\( U - A \)[/tex], is the set of all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]. Therefore, the complement of [tex]\( A \)[/tex] is:
[tex]\[
A' = \{1, 3, 5, 7, 9, 10\}
\][/tex]
Thus, the complement of [tex]\( A = \{2, 4, 6, 8\} \)[/tex] in the universal set [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex] is [tex]\(\{1, 3, 5, 7, 9, 10\}\)[/tex].
