IDNLearn.com: Your trusted source for finding accurate and reliable answers. Our experts provide timely, comprehensive responses to ensure you have the information you need.
Sagot :
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]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. For trustworthy and accurate answers, visit IDNLearn.com. Thanks for stopping by, and see you next time for more solutions.