To find the complement of the set [tex]\( A \)[/tex] in the universe [tex]\( U \)[/tex], follow these steps:
1. Identify the universe set [tex]\( U \)[/tex]: This set contains all the elements under consideration. In this case, the universe is given as:
[tex]\[ U = \{a, b, c, 1, 2, 3\} \][/tex]
2. Identify the subset [tex]\( A \)[/tex]: This is the set for which we need to find the complement. The subset is:
[tex]\[ A = \{b, c, 2\} \][/tex]
3. Understand the concept of the complement: The complement of [tex]\( A \)[/tex] (denoted as [tex]\( A' \)[/tex] or [tex]\( A^c \)[/tex]) in the universe [tex]\( U \)[/tex] consists of all elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
4. Determine the elements of the complement:
- Start with the universe set [tex]\( U = \{a, b, c, 1, 2, 3\} \)[/tex].
- Remove the elements that are in set [tex]\( A = \{b, c, 2\} \)[/tex].
By removing [tex]\( b \)[/tex], [tex]\( c \)[/tex], and [tex]\( 2 \)[/tex] from [tex]\( U \)[/tex]:
[tex]\[
U - A = \{a, b, c, 1, 2, 3\} - \{b, c, 2\} = \{a, 1, 3\}
\][/tex]
5. Write the result:
[tex]\[
A' = \{a, 1, 3\}
\][/tex]
Thus, the complement of the set [tex]\( A \)[/tex] in the universe [tex]\( U \)[/tex] is:
[tex]\[
A' = \{a, 1, 3\}
\][/tex]
