To determine the complement of the set [tex]\( F \)[/tex] in the universal set [tex]\( U \)[/tex], denoted as [tex]\( F' \)[/tex], follow these steps:
1. Identify the Universal Set [tex]\( U \)[/tex]: [tex]\( U = \{2, 4, 6, 8, 10\} \)[/tex]
2. Identify the Set [tex]\( F \)[/tex]: [tex]\( F = \{4, 10\} \)[/tex]
3. Determine the Complement Set [tex]\( F' \)[/tex]: The complement of [tex]\( F \)[/tex], [tex]\( F' \)[/tex], consists of all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( F \)[/tex].
We systematically compare each element of [tex]\( U \)[/tex] to see if it is in [tex]\( F \)[/tex]:
- [tex]\( 2 \)[/tex] is in [tex]\( U \)[/tex] and not in [tex]\( F \)[/tex]; include [tex]\( 2 \)[/tex] in [tex]\( F' \)[/tex]
- [tex]\( 4 \)[/tex] is in [tex]\( U \)[/tex] and is in [tex]\( F \)[/tex]; do not include [tex]\( 4 \)[/tex] in [tex]\( F' \)[/tex]
- [tex]\( 6 \)[/tex] is in [tex]\( U \)[/tex] and not in [tex]\( F \)[/tex]; include [tex]\( 6 \)[/tex] in [tex]\( F' \)[/tex]
- [tex]\( 8 \)[/tex] is in [tex]\( U \)[/tex] and not in [tex]\( F \)[/tex]; include [tex]\( 8 \)[/tex] in [tex]\( F' \)[/tex]
- [tex]\( 10 \)[/tex] is in [tex]\( U \)[/tex] and is in [tex]\( F \)[/tex]; do not include [tex]\( 10 \)[/tex] in [tex]\( F' \)[/tex]
Thus, the set [tex]\( F' \)[/tex], the complement of [tex]\( F \)[/tex] in [tex]\( U \)[/tex], includes [tex]\( 2, 6, \)[/tex] and [tex]\( 8 \)[/tex].
Therefore, the set that represents [tex]\( F' \)[/tex] is:
[tex]\[ \{2, 6, 8\} \][/tex]
Among the provided options, this corresponds to:
[tex]\[ \{2, 6, 8\} \][/tex]
Hence, the correct answer is:
[tex]\[ \{2, 6, 8\} \][/tex]
