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Sagot :
Let's solve the given problem step-by-step.
### Step-by-Step Solution
#### (a) List all the elements of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
1. Universal set [tex]\( \mu \)[/tex]:
[tex]\( \mu = \{ x \, | \, 1 \leq x \leq 18 \} \)[/tex]
Therefore, [tex]\( \mu = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 \} \)[/tex].
2. Set [tex]\( A \)[/tex]: Prime numbers from 1 to 18.
The prime numbers in the given range are [tex]\( \{ 2, 3, 5, 7, 11, 13, 17 \} \)[/tex].
3. Set [tex]\( B \)[/tex]: Odd numbers greater than 3 within the given range.
The odd numbers greater than 3 are [tex]\( \{ 5, 7, 9, 11, 13, 15, 17 \} \)[/tex].
So, the elements of the sets are:
- [tex]\( A = \{ 2, 3, 5, 7, 11, 13, 17 \} \)[/tex]
- [tex]\( B = \{ 5, 7, 9, 11, 13, 15, 17 \} \)[/tex]
#### (b) Find [tex]\( A \cap B \)[/tex]
The intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] consists of the elements that are common to both sets.
- [tex]\( A \cap B \)[/tex] means the elements that are both prime numbers and odd numbers greater than 3.
From the elements of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( A \cap B = \{ 5, 7, 11, 13, 17 \} \)[/tex]
#### (c) Illustrate [tex]\( \mu, A \)[/tex], and [tex]\( B \)[/tex] in a Venn Diagram
Here is a description of how the Venn diagram would be organized:
- Draw a rectangle to represent the universal set [tex]\( \mu \)[/tex].
- Draw two overlapping circles inside the rectangle, one for set [tex]\( A \)[/tex] and the other for set [tex]\( B \)[/tex].
- Place the elements [tex]\( \{ 5, 7, 11, 13, 17 \} \)[/tex] in the overlapping region of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- Place the remaining elements of [tex]\( A \)[/tex] (i.e., [tex]\( \{ 2, 3 \} \)[/tex]) in the part of circle [tex]\( A \)[/tex] that does not overlap with circle [tex]\( B \)[/tex].
- Place the remaining elements of [tex]\( B \)[/tex] (i.e., [tex]\( \{ 9, 15 \} \)[/tex]) in the part of circle [tex]\( B \)[/tex] that does not overlap with circle [tex]\( A \)[/tex].
- Place the elements [tex]\( \{ 1, 4, 6, 8, 10, 12, 14, 16, 18 \} \)[/tex], which are in [tex]\( \mu \)[/tex] but not in [tex]\( A \)[/tex] or [tex]\( B \)[/tex], outside the circles representing [tex]\( A \)[/tex] and [tex]\( B \)[/tex], but inside the rectangle.
#### (d) Shade the region for prime factors of 18 on the Venn diagram
The prime factors of 18 are the prime numbers that multiply together to give 18:
- The prime factorization of 18 is [tex]\( 2 \times 3 \times 3 \)[/tex].
The relevant prime factors of 18 in the context of our sets are [tex]\( 2 \)[/tex] and [tex]\( 3 \)[/tex].
To shade this region on the Venn diagram:
- Locate the positions of [tex]\( 2 \)[/tex] and [tex]\( 3 \)[/tex] in the diagram.
- [tex]\( 2 \)[/tex] is in circle [tex]\( A \)[/tex] but not in circle [tex]\( B \)[/tex].
- [tex]\( 3 \)[/tex] is also in circle [tex]\( A \)[/tex] but outside the overlap with circle [tex]\( B \)[/tex].
Shade the areas around [tex]\( 2 \)[/tex] and [tex]\( 3 \)[/tex] within set [tex]\( A \)[/tex].
### Summary:
- Elements of [tex]\( A \)[/tex]: \{2, 3, 5, 7, 11, 13, 17\}
- Elements of [tex]\( B \)[/tex]: \{5, 7, 9, 11, 13, 15, 17\}
- Intersection [tex]\( A \cap B \)[/tex]: \{5, 7, 11, 13, 17\}
- Prime factors of 18 to be shaded in [tex]\( A \)[/tex]: [tex]\( 2 \)[/tex] and [tex]\( 3 \)[/tex]
Ensure to illustrate [tex]\( \mu, A \)[/tex], and [tex]\( B \)[/tex] as well as shading the prime factors 2 and 3 on your Venn diagram.
### Step-by-Step Solution
#### (a) List all the elements of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]
1. Universal set [tex]\( \mu \)[/tex]:
[tex]\( \mu = \{ x \, | \, 1 \leq x \leq 18 \} \)[/tex]
Therefore, [tex]\( \mu = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 \} \)[/tex].
2. Set [tex]\( A \)[/tex]: Prime numbers from 1 to 18.
The prime numbers in the given range are [tex]\( \{ 2, 3, 5, 7, 11, 13, 17 \} \)[/tex].
3. Set [tex]\( B \)[/tex]: Odd numbers greater than 3 within the given range.
The odd numbers greater than 3 are [tex]\( \{ 5, 7, 9, 11, 13, 15, 17 \} \)[/tex].
So, the elements of the sets are:
- [tex]\( A = \{ 2, 3, 5, 7, 11, 13, 17 \} \)[/tex]
- [tex]\( B = \{ 5, 7, 9, 11, 13, 15, 17 \} \)[/tex]
#### (b) Find [tex]\( A \cap B \)[/tex]
The intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] consists of the elements that are common to both sets.
- [tex]\( A \cap B \)[/tex] means the elements that are both prime numbers and odd numbers greater than 3.
From the elements of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- [tex]\( A \cap B = \{ 5, 7, 11, 13, 17 \} \)[/tex]
#### (c) Illustrate [tex]\( \mu, A \)[/tex], and [tex]\( B \)[/tex] in a Venn Diagram
Here is a description of how the Venn diagram would be organized:
- Draw a rectangle to represent the universal set [tex]\( \mu \)[/tex].
- Draw two overlapping circles inside the rectangle, one for set [tex]\( A \)[/tex] and the other for set [tex]\( B \)[/tex].
- Place the elements [tex]\( \{ 5, 7, 11, 13, 17 \} \)[/tex] in the overlapping region of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- Place the remaining elements of [tex]\( A \)[/tex] (i.e., [tex]\( \{ 2, 3 \} \)[/tex]) in the part of circle [tex]\( A \)[/tex] that does not overlap with circle [tex]\( B \)[/tex].
- Place the remaining elements of [tex]\( B \)[/tex] (i.e., [tex]\( \{ 9, 15 \} \)[/tex]) in the part of circle [tex]\( B \)[/tex] that does not overlap with circle [tex]\( A \)[/tex].
- Place the elements [tex]\( \{ 1, 4, 6, 8, 10, 12, 14, 16, 18 \} \)[/tex], which are in [tex]\( \mu \)[/tex] but not in [tex]\( A \)[/tex] or [tex]\( B \)[/tex], outside the circles representing [tex]\( A \)[/tex] and [tex]\( B \)[/tex], but inside the rectangle.
#### (d) Shade the region for prime factors of 18 on the Venn diagram
The prime factors of 18 are the prime numbers that multiply together to give 18:
- The prime factorization of 18 is [tex]\( 2 \times 3 \times 3 \)[/tex].
The relevant prime factors of 18 in the context of our sets are [tex]\( 2 \)[/tex] and [tex]\( 3 \)[/tex].
To shade this region on the Venn diagram:
- Locate the positions of [tex]\( 2 \)[/tex] and [tex]\( 3 \)[/tex] in the diagram.
- [tex]\( 2 \)[/tex] is in circle [tex]\( A \)[/tex] but not in circle [tex]\( B \)[/tex].
- [tex]\( 3 \)[/tex] is also in circle [tex]\( A \)[/tex] but outside the overlap with circle [tex]\( B \)[/tex].
Shade the areas around [tex]\( 2 \)[/tex] and [tex]\( 3 \)[/tex] within set [tex]\( A \)[/tex].
### Summary:
- Elements of [tex]\( A \)[/tex]: \{2, 3, 5, 7, 11, 13, 17\}
- Elements of [tex]\( B \)[/tex]: \{5, 7, 9, 11, 13, 15, 17\}
- Intersection [tex]\( A \cap B \)[/tex]: \{5, 7, 11, 13, 17\}
- Prime factors of 18 to be shaded in [tex]\( A \)[/tex]: [tex]\( 2 \)[/tex] and [tex]\( 3 \)[/tex]
Ensure to illustrate [tex]\( \mu, A \)[/tex], and [tex]\( B \)[/tex] as well as shading the prime factors 2 and 3 on your Venn diagram.
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