Answer :
Let's solve the given problem step-by-step:
### Given Sets:
- Universal set [tex]\( U = \{m, q, r, s, x, y, z\} \)[/tex]
- Set [tex]\( B = \{r, x, z\} \)[/tex]
- Set [tex]\( D = \{x, y\} \)[/tex]
We need to find:
(a) [tex]\( B \cap D^{\prime} \)[/tex]
(b) [tex]\( (B \cup D)^{\prime} \)[/tex]
### Part (a)
To find [tex]\( B \cap D^{\prime} \)[/tex]:
1. Complement of [tex]\( D \)[/tex] (denoted as [tex]\( D^{\prime} \)[/tex]):
[tex]\( D^{\prime} = U - D \)[/tex].
Subtract the elements of [tex]\( D \)[/tex] from [tex]\( U \)[/tex].
[tex]\[ D^{\prime} = \{m, q, r, s, z\} \][/tex]
2. Intersection of [tex]\( B \)[/tex] and [tex]\( D^{\prime} \)[/tex]:
[tex]\( B \cap D^{\prime} \)[/tex] is the set of elements that are in both [tex]\( B \)[/tex] and [tex]\( D^{\prime} \)[/tex].
[tex]\[ B = \{r, x, z\} \][/tex]
[tex]\[ D^{\prime} = \{m, q, r, s, z\} \][/tex]
Find the common elements between [tex]\( B \)[/tex] and [tex]\( D^{\prime} \)[/tex]:
[tex]\[ B \cap D^{\prime} = \{r, z\} \][/tex]
3. Answer for part (a):
[tex]\[ B \cap D^{\prime} = \{r, z\} \][/tex]
### Part (b)
To find [tex]\( (B \cup D)^{\prime} \)[/tex]:
1. Union of [tex]\( B \)[/tex] and [tex]\( D \)[/tex] (denoted as [tex]\( B \cup D \)[/tex]):
[tex]\( B \cup D \)[/tex] is the set of all elements that are in [tex]\( B \)[/tex], or [tex]\( D \)[/tex], or both.
[tex]\[ B = \{r, x, z\} \][/tex]
[tex]\[ D = \{x, y\} \][/tex]
Combine the elements of [tex]\( B \)[/tex] and [tex]\( D \)[/tex]:
[tex]\[ B \cup D = \{r, x, z, y\} \][/tex]
2. Complement of [tex]\( B \cup D \)[/tex] (denoted as [tex]\( (B \cup D)^{\prime} \)[/tex]):
[tex]\( (B \cup D)^{\prime} = U - (B \cup D) \)[/tex].
Subtract the elements of [tex]\( B \cup D \)[/tex] from [tex]\( U \)[/tex]:
[tex]\[ B \cup D = \{r, x, z, y\} \][/tex]
[tex]\[ U = \{m, q, r, s, x, y, z\} \][/tex]
[tex]\[ (B \cup D)^{\prime} = \{m, q, s\} \][/tex]
3. Answer for part (b):
[tex]\[ (B \cup D)^{\prime} = \{m, q, s\} \][/tex]
### Final Answers:
(a) [tex]\( B \cap D^{\prime} = \{r, z\} \)[/tex]
(b) [tex]\( (B \cup D)^{\prime} = \{m, q, s\} \)[/tex]
### Given Sets:
- Universal set [tex]\( U = \{m, q, r, s, x, y, z\} \)[/tex]
- Set [tex]\( B = \{r, x, z\} \)[/tex]
- Set [tex]\( D = \{x, y\} \)[/tex]
We need to find:
(a) [tex]\( B \cap D^{\prime} \)[/tex]
(b) [tex]\( (B \cup D)^{\prime} \)[/tex]
### Part (a)
To find [tex]\( B \cap D^{\prime} \)[/tex]:
1. Complement of [tex]\( D \)[/tex] (denoted as [tex]\( D^{\prime} \)[/tex]):
[tex]\( D^{\prime} = U - D \)[/tex].
Subtract the elements of [tex]\( D \)[/tex] from [tex]\( U \)[/tex].
[tex]\[ D^{\prime} = \{m, q, r, s, z\} \][/tex]
2. Intersection of [tex]\( B \)[/tex] and [tex]\( D^{\prime} \)[/tex]:
[tex]\( B \cap D^{\prime} \)[/tex] is the set of elements that are in both [tex]\( B \)[/tex] and [tex]\( D^{\prime} \)[/tex].
[tex]\[ B = \{r, x, z\} \][/tex]
[tex]\[ D^{\prime} = \{m, q, r, s, z\} \][/tex]
Find the common elements between [tex]\( B \)[/tex] and [tex]\( D^{\prime} \)[/tex]:
[tex]\[ B \cap D^{\prime} = \{r, z\} \][/tex]
3. Answer for part (a):
[tex]\[ B \cap D^{\prime} = \{r, z\} \][/tex]
### Part (b)
To find [tex]\( (B \cup D)^{\prime} \)[/tex]:
1. Union of [tex]\( B \)[/tex] and [tex]\( D \)[/tex] (denoted as [tex]\( B \cup D \)[/tex]):
[tex]\( B \cup D \)[/tex] is the set of all elements that are in [tex]\( B \)[/tex], or [tex]\( D \)[/tex], or both.
[tex]\[ B = \{r, x, z\} \][/tex]
[tex]\[ D = \{x, y\} \][/tex]
Combine the elements of [tex]\( B \)[/tex] and [tex]\( D \)[/tex]:
[tex]\[ B \cup D = \{r, x, z, y\} \][/tex]
2. Complement of [tex]\( B \cup D \)[/tex] (denoted as [tex]\( (B \cup D)^{\prime} \)[/tex]):
[tex]\( (B \cup D)^{\prime} = U - (B \cup D) \)[/tex].
Subtract the elements of [tex]\( B \cup D \)[/tex] from [tex]\( U \)[/tex]:
[tex]\[ B \cup D = \{r, x, z, y\} \][/tex]
[tex]\[ U = \{m, q, r, s, x, y, z\} \][/tex]
[tex]\[ (B \cup D)^{\prime} = \{m, q, s\} \][/tex]
3. Answer for part (b):
[tex]\[ (B \cup D)^{\prime} = \{m, q, s\} \][/tex]
### Final Answers:
(a) [tex]\( B \cap D^{\prime} = \{r, z\} \)[/tex]
(b) [tex]\( (B \cup D)^{\prime} = \{m, q, s\} \)[/tex]
