Let's break down the problem step by step to find the required sets.
### Given:
- [tex]\(U = \{1, 2, 3, 4, 6, 7, 8\}\)[/tex]
- [tex]\(B = \{2, 7, 8\}\)[/tex]
- [tex]\(C = \{2, 4, 6, 7\}\)[/tex]
### Step-by-step Solution:
#### Part (a): Find [tex]\(B \cup C^{\prime}\)[/tex]
1. Find [tex]\(C^{\prime}\)[/tex]: The complement of set [tex]\(C\)[/tex] includes all elements in the universal set [tex]\(U\)[/tex] that are not in [tex]\(C\)[/tex].
[tex]\[
C^{\prime} = U - C = \{1, 3, 8\} - \{2, 4, 6, 7\} = \{1, 3, 8\}
\][/tex]
2. Find [tex]\(B \cup C^{\prime}\)[/tex]: The union of sets [tex]\(B\)[/tex] and [tex]\(C^{\prime}\)[/tex] includes all elements that are in either [tex]\(B\)[/tex] or [tex]\(C^{\prime}\)[/tex].
[tex]\[
B \cup C^{\prime} = \{2, 7, 8\} \cup \{1, 3, 8\} = \{1, 2, 3, 7, 8\}
\][/tex]
So, [tex]\(B \cup C^{\prime} = \{1, 2, 3, 7, 8\}\)[/tex].
#### Part (b): Find [tex]\(B^{\prime} \cap C^{\prime}\)[/tex]
1. Find [tex]\(B^{\prime}\)[/tex]: The complement of set [tex]\(B\)[/tex] includes all elements in the universal set [tex]\(U\)[/tex] that are not in [tex]\(B\)[/tex].
[tex]\[
B^{\prime} = U - B = \{1, 3, 4, 6\}
\][/tex]
2. Find [tex]\(B^{\prime} \cap C^{\prime}\)[/tex]: The intersection of sets [tex]\(B^{\prime}\)[/tex] and [tex]\(C^{\prime}\)[/tex] includes all elements that are in both [tex]\(B^{\prime}\)[/tex] and [tex]\(C^{\prime}\)[/tex].
[tex]\[
B^{\prime} \cap C^{\prime} = \{1, 3, 4, 6\} \cap \{1, 3, 8\} = \{1, 3\}
\][/tex]
So, [tex]\(B^{\prime} \cap C^{\prime} = \{1, 3\}\)[/tex].
### Final Answer:
(a) [tex]\(B \cup C^{\prime} = \{1, 2, 3, 7, 8\}\)[/tex]
(b) [tex]\(B^{\prime} \cap C^{\prime} = \{1, 3\}\)[/tex]
