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Sets [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are subsets of the universal set [tex]\( U \)[/tex].
These sets are defined as follows:

[tex]\[
\begin{array}{l}
U = \{1, 2, 3, 4, 6, 7, 8\} \\
B = \{2, 7, 8\} \\
C = \{2, 4, 6, 7\}
\end{array}
\][/tex]

Find the following sets.
Write your answer in roster form or as [tex]\( \varnothing \)[/tex].

(a) [tex]\( B \cup C' = \square \)[/tex]

(b) [tex]\( B' \cap C' = \square \)[/tex]

Sagot :

GinnyAnsweridn
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]
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