Answer :
Sure, let's break down each part of the question step-by-step using the given sets [tex]\( U \)[/tex], [tex]\( A \)[/tex], and [tex]\( B \)[/tex].
### (i) Verification: Given Sets
We have the universal set [tex]\( U \)[/tex], set [tex]\( A \)[/tex], and set [tex]\( B \)[/tex] defined as:
- [tex]\( U = \{1, 2, 3, \ldots, 9\} \)[/tex]
- [tex]\( A = \{1, 2, 3, 4, 6, 7, 8\} \)[/tex]
- [tex]\( B = \{4, 6, 8\} \)[/tex]
Let's verify the different operations on these sets.
#### a) Union of A and B: [tex]\( A \cup B \)[/tex]
The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is defined as the set of elements that are in [tex]\( A \)[/tex], or [tex]\( B \)[/tex], or in both.
[tex]\( A \cup B = \{1, 2, 3, 4, 6, 7, 8\} \cup \{4, 6, 8\} \)[/tex]
Combining all unique elements from both sets, we get:
[tex]\[ A \cup B = \{1, 2, 3, 4, 6, 7, 8\} \][/tex]
This verifies that the union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( \{1, 2, 3, 4, 6, 7, 8\} \)[/tex].
#### b) Intersection of A and B: [tex]\( A \cap B \)[/tex]
The intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is defined as the set of elements that are common to both [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
[tex]\( A \cap B = \{1, 2, 3, 4, 6, 7, 8\} \cap \{4, 6, 8\} \)[/tex]
Identifying common elements, we get:
[tex]\[ A \cap B = \{4, 6, 8\} \][/tex]
This verifies that the intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( \{4, 6, 8\} \)[/tex].
#### c) Complement of A: [tex]\( A^c \)[/tex]
The complement of set [tex]\( A \)[/tex] in the universal set [tex]\( U \)[/tex] is defined as the set of elements that are in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
[tex]\( A^c = U - A = \{1, 2, 3, \ldots, 9\} - \{1, 2, 3, 4, 6, 7, 8\} \)[/tex]
Removing all elements of [tex]\( A \)[/tex] from [tex]\( U \)[/tex], we are left with:
[tex]\[ A^c = \{9, 5\} \][/tex]
This verifies that the complement of [tex]\( A \)[/tex] is [tex]\( \{9, 5\} \)[/tex].
#### d) Complement of B: [tex]\( B^c \)[/tex]
The complement of set [tex]\( B \)[/tex] in the universal set [tex]\( U \)[/tex] is defined as the set of elements that are in [tex]\( U \)[/tex] but not in [tex]\( B \)[/tex].
[tex]\( B^c = U - B = \{1, 2, 3, \ldots, 9\} - \{4, 6, 8\} \)[/tex]
Removing all elements of [tex]\( B \)[/tex] from [tex]\( U \)[/tex], we are left with:
[tex]\[ B^c = \{1, 2, 3, 5, 7, 9\} \][/tex]
This verifies that the complement of [tex]\( B \)[/tex] is [tex]\( \{1, 2, 3, 5, 7, 9\} \)[/tex].
By verifying each operation on the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] with respect to the universal set [tex]\( U \)[/tex], we can confirm that:
- [tex]\( A \cup B = \{1, 2, 3, 4, 6, 7, 8\} \)[/tex]
- [tex]\( A \cap B = \{4, 6, 8\} \)[/tex]
- [tex]\( A^c = \{9, 5\} \)[/tex]
- [tex]\( B^c = \{1, 2, 3, 5, 7, 9\} \)[/tex]
### (i) Verification: Given Sets
We have the universal set [tex]\( U \)[/tex], set [tex]\( A \)[/tex], and set [tex]\( B \)[/tex] defined as:
- [tex]\( U = \{1, 2, 3, \ldots, 9\} \)[/tex]
- [tex]\( A = \{1, 2, 3, 4, 6, 7, 8\} \)[/tex]
- [tex]\( B = \{4, 6, 8\} \)[/tex]
Let's verify the different operations on these sets.
#### a) Union of A and B: [tex]\( A \cup B \)[/tex]
The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is defined as the set of elements that are in [tex]\( A \)[/tex], or [tex]\( B \)[/tex], or in both.
[tex]\( A \cup B = \{1, 2, 3, 4, 6, 7, 8\} \cup \{4, 6, 8\} \)[/tex]
Combining all unique elements from both sets, we get:
[tex]\[ A \cup B = \{1, 2, 3, 4, 6, 7, 8\} \][/tex]
This verifies that the union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( \{1, 2, 3, 4, 6, 7, 8\} \)[/tex].
#### b) Intersection of A and B: [tex]\( A \cap B \)[/tex]
The intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is defined as the set of elements that are common to both [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
[tex]\( A \cap B = \{1, 2, 3, 4, 6, 7, 8\} \cap \{4, 6, 8\} \)[/tex]
Identifying common elements, we get:
[tex]\[ A \cap B = \{4, 6, 8\} \][/tex]
This verifies that the intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( \{4, 6, 8\} \)[/tex].
#### c) Complement of A: [tex]\( A^c \)[/tex]
The complement of set [tex]\( A \)[/tex] in the universal set [tex]\( U \)[/tex] is defined as the set of elements that are in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex].
[tex]\( A^c = U - A = \{1, 2, 3, \ldots, 9\} - \{1, 2, 3, 4, 6, 7, 8\} \)[/tex]
Removing all elements of [tex]\( A \)[/tex] from [tex]\( U \)[/tex], we are left with:
[tex]\[ A^c = \{9, 5\} \][/tex]
This verifies that the complement of [tex]\( A \)[/tex] is [tex]\( \{9, 5\} \)[/tex].
#### d) Complement of B: [tex]\( B^c \)[/tex]
The complement of set [tex]\( B \)[/tex] in the universal set [tex]\( U \)[/tex] is defined as the set of elements that are in [tex]\( U \)[/tex] but not in [tex]\( B \)[/tex].
[tex]\( B^c = U - B = \{1, 2, 3, \ldots, 9\} - \{4, 6, 8\} \)[/tex]
Removing all elements of [tex]\( B \)[/tex] from [tex]\( U \)[/tex], we are left with:
[tex]\[ B^c = \{1, 2, 3, 5, 7, 9\} \][/tex]
This verifies that the complement of [tex]\( B \)[/tex] is [tex]\( \{1, 2, 3, 5, 7, 9\} \)[/tex].
By verifying each operation on the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] with respect to the universal set [tex]\( U \)[/tex], we can confirm that:
- [tex]\( A \cup B = \{1, 2, 3, 4, 6, 7, 8\} \)[/tex]
- [tex]\( A \cap B = \{4, 6, 8\} \)[/tex]
- [tex]\( A^c = \{9, 5\} \)[/tex]
- [tex]\( B^c = \{1, 2, 3, 5, 7, 9\} \)[/tex]
