Let's solve each part of the question step-by-step.
### Given sets:
- [tex]\( U = \{1, 2, 3, 7, 8, 9\} \)[/tex]
- [tex]\( B = \{1, 2, 3\} \)[/tex]
- [tex]\( D = \{2, 3, 7\} \)[/tex]
### Part (a): [tex]\((B \cap D)^{\prime}\)[/tex]
1. Find [tex]\( B \cap D \)[/tex]:
- Intersection of [tex]\( B \)[/tex] and [tex]\( D \)[/tex] includes elements that are in both sets.
- [tex]\( B \cap D = \{2, 3\} \)[/tex]
2. Find the complement of [tex]\( B \cap D \)[/tex] ([tex]\((B \cap D)^{\prime}\)[/tex]):
- The complement of a set includes elements that are in the universal set [tex]\( U \)[/tex] but not in the intersected set.
- [tex]\( (B \cap D)^{\prime} = U - (B \cap D) \)[/tex]
- Therefore, [tex]\( (B \cap D)^{\prime} = \{1, 2, 3, 7, 8, 9\} - \{2, 3\} \)[/tex]
- [tex]\( (B \cap D)^{\prime} = \{1, 7, 8, 9\} \)[/tex]
[tex]\[ (B \cap D)^{\prime} = \{1, 7, 8, 9\} \][/tex]
### Part (b): [tex]\( B^{\prime} \cup D \)[/tex]
1. Find [tex]\( B^{\prime} \)[/tex]:
- The complement of [tex]\( B \)[/tex] includes elements that are in the universal set [tex]\( U \)[/tex] but not in [tex]\( B \)[/tex].
- [tex]\( B^{\prime} = U - B \)[/tex]
- Therefore, [tex]\( B^{\prime} = \{1, 2, 3, 7, 8, 9\} - \{1, 2, 3\} \)[/tex]
- [tex]\( B^{\prime} = \{7, 8, 9\} \)[/tex]
2. Find the union of [tex]\( B^{\prime} \cup D \)[/tex]:
- The union of two sets includes all elements that are in either of the sets or both.
- [tex]\( B^{\prime} \cup D = \{7, 8, 9\} \cup \{2, 3, 7\} \)[/tex]
- Combine all unique elements from both sets.
- [tex]\( B^{\prime} \cup D = \{2, 3, 7, 8, 9\} \)[/tex]
[tex]\[ B^{\prime} \cup D = \{2, 3, 7, 8, 9\} \][/tex]
### Final Answers:
(a) [tex]\((B \cap D)^{\prime} = \{1, 7, 8, 9\}\)[/tex]
(b) [tex]\(B^{\prime} \cup D = \{2, 3, 7, 8, 9\}\)[/tex]
