Answer :
Let's break down the problem step-by-step to find the required sets and their complements and intersections.
### Step 1: List the members of [tex]\( U \)[/tex], set [tex]\( A \)[/tex], and set [tex]\( B \)[/tex].
Universal Set [tex]\( U \)[/tex]:
The set [tex]\( U \)[/tex] contains all positive integers not greater than 9:
[tex]\[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \][/tex]
Set [tex]\( A \)[/tex]:
Set [tex]\( A \)[/tex] contains all positive odd numbers less than 13:
[tex]\[ A = \{1, 3, 5, 7, 9, 11, 13\} \][/tex]
Set [tex]\( B \)[/tex]:
Set [tex]\( B \)[/tex] contains all positive integers that are factors of 3 in the range of the universal set [tex]\( U \)[/tex]:
[tex]\[ B = \{3, 6, 9\} \][/tex]
### Step 2: Draw a diagram to represent [tex]\( U \)[/tex], set [tex]\( A \)[/tex], and set [tex]\( B \)[/tex].
To draw a Venn diagram:
1. Draw a rectangle to represent the universal set [tex]\( U \)[/tex].
2. Inside this rectangle, draw two overlapping circles to represent sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Since I cannot draw here, visualize the rectangle containing [tex]\( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)[/tex]. Inside this rectangle, draw two circles such that:
- The left circle represents [tex]\( A \)[/tex] and contains [tex]\( \{1, 3, 5, 7, 9 \} \)[/tex].
- The right circle represents [tex]\( B \)[/tex] and contains [tex]\( \{3, 6, 9 \} \)[/tex].
- The overlap (intersection) of the two circles contains [tex]\( \{3, 9 \} \)[/tex].
### Step 3: Find [tex]\( (A \cup B)^{\prime} \)[/tex] and [tex]\( A \cap B^{\prime} \)[/tex].
Step 3.1: Find [tex]\( A \cup B \)[/tex]:
[tex]\[ A \cup B = \{1, 3, 5, 7, 9, 11, 13\} \cup \{3, 6, 9\} = \{1, 3, 5, 6, 7, 9, 11, 13\} \][/tex]
However, since we are constrained by the universal set [tex]\( U \)[/tex], only consider elements within [tex]\( U \)[/tex]:
[tex]\[ A \cup B \cap U = \{1, 3, 5, 6, 7, 9\} \][/tex]
Step 3.2: Find [tex]\( (A \cup B)^{\prime} \)[/tex]:
To find [tex]\( (A \cup B)^{\prime} \)[/tex], we take the complement relative to the universal set [tex]\( U \)[/tex]:
[tex]\[ (A \cup B)^{\prime} = U - (A \cup B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{1, 3, 5, 6, 7, 9\} = \{2, 4, 8\} \][/tex]
Step 3.3: Find [tex]\( B^{\prime} \)[/tex]:
To find the complement of [tex]\( B \)[/tex] relative to [tex]\( U \)[/tex]:
[tex]\[ B^{\prime} = U - B = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{3, 6, 9\} = \{1, 2, 4, 5, 7, 8\} \][/tex]
Step 3.4: Find [tex]\( A \cap B^{\prime} \)[/tex]:
We intersect [tex]\( A \)[/tex] with [tex]\( B^{\prime} \)[/tex] to find the elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]:
[tex]\[ A \cap B^{\prime} = \{1, 3, 5, 6, 7, 9, 11, 13\} \cap \{1, 2, 4, 5, 7, 8\} = \{1, 5, 7\} \][/tex]
### Summary:
1. [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)[/tex]
2. [tex]\( A = \{1, 3, 5, 7, 9, 11, 13\} \)[/tex]
3. [tex]\( B = \{3, 6, 9\} \)[/tex]
4. [tex]\( (A \cup B)^{\prime} = \{2, 4, 8\} \)[/tex]
5. [tex]\( A \cap B^{\prime} = \{1, 5, 7\} \)[/tex]
### Step 1: List the members of [tex]\( U \)[/tex], set [tex]\( A \)[/tex], and set [tex]\( B \)[/tex].
Universal Set [tex]\( U \)[/tex]:
The set [tex]\( U \)[/tex] contains all positive integers not greater than 9:
[tex]\[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \][/tex]
Set [tex]\( A \)[/tex]:
Set [tex]\( A \)[/tex] contains all positive odd numbers less than 13:
[tex]\[ A = \{1, 3, 5, 7, 9, 11, 13\} \][/tex]
Set [tex]\( B \)[/tex]:
Set [tex]\( B \)[/tex] contains all positive integers that are factors of 3 in the range of the universal set [tex]\( U \)[/tex]:
[tex]\[ B = \{3, 6, 9\} \][/tex]
### Step 2: Draw a diagram to represent [tex]\( U \)[/tex], set [tex]\( A \)[/tex], and set [tex]\( B \)[/tex].
To draw a Venn diagram:
1. Draw a rectangle to represent the universal set [tex]\( U \)[/tex].
2. Inside this rectangle, draw two overlapping circles to represent sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Since I cannot draw here, visualize the rectangle containing [tex]\( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)[/tex]. Inside this rectangle, draw two circles such that:
- The left circle represents [tex]\( A \)[/tex] and contains [tex]\( \{1, 3, 5, 7, 9 \} \)[/tex].
- The right circle represents [tex]\( B \)[/tex] and contains [tex]\( \{3, 6, 9 \} \)[/tex].
- The overlap (intersection) of the two circles contains [tex]\( \{3, 9 \} \)[/tex].
### Step 3: Find [tex]\( (A \cup B)^{\prime} \)[/tex] and [tex]\( A \cap B^{\prime} \)[/tex].
Step 3.1: Find [tex]\( A \cup B \)[/tex]:
[tex]\[ A \cup B = \{1, 3, 5, 7, 9, 11, 13\} \cup \{3, 6, 9\} = \{1, 3, 5, 6, 7, 9, 11, 13\} \][/tex]
However, since we are constrained by the universal set [tex]\( U \)[/tex], only consider elements within [tex]\( U \)[/tex]:
[tex]\[ A \cup B \cap U = \{1, 3, 5, 6, 7, 9\} \][/tex]
Step 3.2: Find [tex]\( (A \cup B)^{\prime} \)[/tex]:
To find [tex]\( (A \cup B)^{\prime} \)[/tex], we take the complement relative to the universal set [tex]\( U \)[/tex]:
[tex]\[ (A \cup B)^{\prime} = U - (A \cup B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{1, 3, 5, 6, 7, 9\} = \{2, 4, 8\} \][/tex]
Step 3.3: Find [tex]\( B^{\prime} \)[/tex]:
To find the complement of [tex]\( B \)[/tex] relative to [tex]\( U \)[/tex]:
[tex]\[ B^{\prime} = U - B = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{3, 6, 9\} = \{1, 2, 4, 5, 7, 8\} \][/tex]
Step 3.4: Find [tex]\( A \cap B^{\prime} \)[/tex]:
We intersect [tex]\( A \)[/tex] with [tex]\( B^{\prime} \)[/tex] to find the elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]:
[tex]\[ A \cap B^{\prime} = \{1, 3, 5, 6, 7, 9, 11, 13\} \cap \{1, 2, 4, 5, 7, 8\} = \{1, 5, 7\} \][/tex]
### Summary:
1. [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)[/tex]
2. [tex]\( A = \{1, 3, 5, 7, 9, 11, 13\} \)[/tex]
3. [tex]\( B = \{3, 6, 9\} \)[/tex]
4. [tex]\( (A \cup B)^{\prime} = \{2, 4, 8\} \)[/tex]
5. [tex]\( A \cap B^{\prime} = \{1, 5, 7\} \)[/tex]
