Let's solve each of the given sets step-by-step.
Given sets:
- The universal set [tex]\( U = \{1,2,3,4,5,6,7,8\} \)[/tex]
- Set [tex]\( A = \{1,3,4,5\} \)[/tex]
- Set [tex]\( B = \{2,4,6,8\} \)[/tex]
### a. [tex]\( A \cup B \)[/tex] (A union B)
The union of two sets, [tex]\( A \cup B \)[/tex], is the set of elements that are in either [tex]\( A \)[/tex], [tex]\( B \)[/tex], or both.
List the elements in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A \cup B = \{1, 2, 3, 4, 5, 6, 8\}
\][/tex]
### b. [tex]\( A \cap B \)[/tex] (A intersection B)
The intersection of two sets, [tex]\( A \cap B \)[/tex], is the set of elements that are common to both [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
List the elements that are common in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A \cap B = \{4\}
\][/tex]
### c. [tex]\( A^{\prime} \)[/tex] (Complement of A)
The complement of set [tex]\( A \)[/tex], denoted as [tex]\( A^{\prime} \)[/tex] or [tex]\( A^c \)[/tex], consists of all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
List the elements in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex]:
[tex]\[
A^{\prime} = \{2, 6, 7, 8\}
\][/tex]
### Summary of Results
a. [tex]\( A \cup B = \{1, 2, 3, 4, 5, 6, 8\} \)[/tex]
b. [tex]\( A \cap B = \{4\} \)[/tex]
c. [tex]\( A^{\prime} = \{2, 6, 7, 8\} \)[/tex]
