Let's solve the given problem step-by-step.
Given sets are:
[tex]\[ U = \{2, 3, 4, 5, 6, 7\} \][/tex]
[tex]\[ B = \{3, 4, 5\} \][/tex]
[tex]\[ C = \{3, 5, 6\} \][/tex]
### Part (a) [tex]\((B \cap C)^{\prime}\)[/tex]
First, we need to find the intersection of sets [tex]\(B\)[/tex] and [tex]\(C\)[/tex]:
[tex]\[ B \cap C = \{x \mid x \in B \text{ and } x \in C\} \][/tex]
Thus:
[tex]\[ B \cap C = \{3, 5\} \][/tex]
Next, we find the complement of [tex]\(B \cap C\)[/tex] in the universal set [tex]\(U\)[/tex]:
[tex]\[ (B \cap C)^{\prime} = U - (B \cap C) \][/tex]
So:
[tex]\[ (B \cap C)^{\prime} = \{2, 3, 4, 5, 6, 7\} - \{3, 5\} = \{2, 4, 6, 7\} \][/tex]
Therefore:
[tex]\[ (B \cap C)^{\prime} = \{2, 4, 6, 7\} \][/tex]
### Part (b) [tex]\(B^{\prime} \cup C\)[/tex]
First, we need to find the complement of set [tex]\(B\)[/tex] in the universal set [tex]\(U\)[/tex]:
[tex]\[ B^{\prime} = U - B \][/tex]
So:
[tex]\[ B^{\prime} = \{2, 3, 4, 5, 6, 7\} - \{3, 4, 5\} = \{2, 6, 7\} \][/tex]
Next, we find the union of [tex]\(B^{\prime}\)[/tex] and [tex]\(C\)[/tex]:
[tex]\[ B^{\prime} \cup C = \{x \mid x \in B^{\prime} \text{ or } x \in C\} \][/tex]
Thus:
[tex]\[ B^{\prime} = \{2, 6, 7\} \][/tex]
[tex]\[ C = \{3, 5, 6\} \][/tex]
The union is:
[tex]\[ B^{\prime} \cup C = \{2, 6, 7\} \cup \{3, 5, 6\} = \{2, 3, 5, 6, 7\} \][/tex]
Therefore:
[tex]\[ B^{\prime} \cup C = \{2, 3, 5, 6, 7\} \][/tex]
In summary:
(a) [tex]\((B \cap C)^{\prime} = \{2, 4, 6, 7\}\)[/tex]
(b) [tex]\(B^{\prime} \cup C = \{2, 3, 5, 6, 7\}\)[/tex]
