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Sagot :
Let's tackle the problem step-by-step:
### Part a)
Determine if Sets X and Y are Equal or Equivalent
1. Definitions:
- Two sets are equal if they have exactly the same elements.
- Two sets are equivalent if they have the same number of elements (i.e., they have the same cardinality).
2. Set X: Prime Numbers Less Than 20
- [tex]\(X = \{2, 3, 5, 7, 11, 13, 17, 19\}\)[/tex]
- Set X includes 8 elements: {2, 3, 5, 7, 11, 13, 17, 19}.
3. Set Y: First Six Multiples of 41
- The first six multiples of 41 are: [tex]\( 41 \times 1, 41 \times 2, 41 \times 3, 41 \times 4, 41 \times 5, 41 \times 6 \)[/tex]
- Computing these, we get: [tex]\( Y = \{41, 82, 123, 164, 205, 246\} \)[/tex]
- Set Y includes 6 elements: {41, 82, 123, 164, 205, 246}.
4. Analysis of Equality:
- Set X includes {2, 3, 5, 7, 11, 13, 17, 19}.
- Set Y includes {41, 82, 123, 164, 205, 246}.
- Since the elements in Set X and Set Y are completely different, we conclude that X and Y are not equal.
5. Analysis of Equivalence:
- The cardinality (number of elements) of Set X is 8.
- The cardinality (number of elements) of Set Y is 6.
- Since the number of elements in Set X is different from the number of elements in Set Y, we conclude that X and Y are not equivalent.
### Conclusion for Part a:
- X and Y are neither equal nor equivalent sets.
### Part b)
Writing Sets in Specified Methods
1. Listing Method for Set X:
- The listing method involves explicitly writing out all the elements of the set.
- [tex]\( X = \{2, 3, 5, 7, 11, 13, 17, 19\} \)[/tex]
2. Set Builder Method for Set Y:
- The set builder method involves describing the properties that characterize the elements of the set.
- [tex]\( Y = \{ 41 \times i \mid i \in \mathbb{N}, 1 \leq i \leq 6 \} \)[/tex]
- This means Y is the set of elements that are multiples of 41, where [tex]\( i \)[/tex] is a natural number between 1 and 6 (inclusive).
### Conclusion for Part b:
- Set X in listing method: [tex]\( X = \{2, 3, 5, 7, 11, 13, 17, 19\} \)[/tex]
- Set Y in set builder method: [tex]\( Y = \{ 41 \times i \mid i \in \mathbb{N}, 1 \leq i \leq 6 \} \)[/tex]
By following this detailed and methodical approach, we have addressed both parts of the problem accurately.
### Part a)
Determine if Sets X and Y are Equal or Equivalent
1. Definitions:
- Two sets are equal if they have exactly the same elements.
- Two sets are equivalent if they have the same number of elements (i.e., they have the same cardinality).
2. Set X: Prime Numbers Less Than 20
- [tex]\(X = \{2, 3, 5, 7, 11, 13, 17, 19\}\)[/tex]
- Set X includes 8 elements: {2, 3, 5, 7, 11, 13, 17, 19}.
3. Set Y: First Six Multiples of 41
- The first six multiples of 41 are: [tex]\( 41 \times 1, 41 \times 2, 41 \times 3, 41 \times 4, 41 \times 5, 41 \times 6 \)[/tex]
- Computing these, we get: [tex]\( Y = \{41, 82, 123, 164, 205, 246\} \)[/tex]
- Set Y includes 6 elements: {41, 82, 123, 164, 205, 246}.
4. Analysis of Equality:
- Set X includes {2, 3, 5, 7, 11, 13, 17, 19}.
- Set Y includes {41, 82, 123, 164, 205, 246}.
- Since the elements in Set X and Set Y are completely different, we conclude that X and Y are not equal.
5. Analysis of Equivalence:
- The cardinality (number of elements) of Set X is 8.
- The cardinality (number of elements) of Set Y is 6.
- Since the number of elements in Set X is different from the number of elements in Set Y, we conclude that X and Y are not equivalent.
### Conclusion for Part a:
- X and Y are neither equal nor equivalent sets.
### Part b)
Writing Sets in Specified Methods
1. Listing Method for Set X:
- The listing method involves explicitly writing out all the elements of the set.
- [tex]\( X = \{2, 3, 5, 7, 11, 13, 17, 19\} \)[/tex]
2. Set Builder Method for Set Y:
- The set builder method involves describing the properties that characterize the elements of the set.
- [tex]\( Y = \{ 41 \times i \mid i \in \mathbb{N}, 1 \leq i \leq 6 \} \)[/tex]
- This means Y is the set of elements that are multiples of 41, where [tex]\( i \)[/tex] is a natural number between 1 and 6 (inclusive).
### Conclusion for Part b:
- Set X in listing method: [tex]\( X = \{2, 3, 5, 7, 11, 13, 17, 19\} \)[/tex]
- Set Y in set builder method: [tex]\( Y = \{ 41 \times i \mid i \in \mathbb{N}, 1 \leq i \leq 6 \} \)[/tex]
By following this detailed and methodical approach, we have addressed both parts of the problem accurately.
### (a) Determine if Sets \( X \) and \( Y \) are Equal or Equivalent
**Equal Sets:**
Two sets are equal if they have exactly the same elements.
**Equivalent Sets:**
Two sets are equivalent if they have the same number of elements, regardless of what those elements are.
#### Given:
- \( X = \) [prime numbers less than 20]
- \( Y = \) [first six multiples of 41]
#### Set \( X \):
Prime numbers less than 20 are: 2, 3, 5, 7, 11, 13, 17, 19.
Thus,
\[ X = \{2, 3, 5, 7, 11, 13, 17, 19\} \]
Number of elements in \( X \): 8.
#### Set \( Y \):
First six multiples of 41 are: 41, 82, 123, 164, 205, 246.
Thus,
\[ Y = \{41, 82, 123, 164, 205, 246\} \]
Number of elements in \( Y \): 6.
**Comparison:**
- The sets \( X \) and \( Y \) do not contain the same elements, so they are not equal.
- The set \( X \) has 8 elements, while the set \( Y \) has 6 elements. Since they do not have the same number of elements, they are not equivalent.
**Conclusion:**
Sets \( X \) and \( Y \) are neither equal nor equivalent.
### (b) Write the Sets Using Different Methods
**Set \( X \) in the Listing Method:**
Set \( X \) is written by listing all its elements explicitly:
\[ X = \{2, 3, 5, 7, 11, 13, 17, 19\} \]
**Set \( Y \) in the Set Builder Method:**
Set \( Y \) can be described as the set of the first six multiples of 41:
\[ Y = \{41n \mid n \in \mathbb{N} \text{ and } 1 \leq n \leq 6\} \]
Here, \(\mathbb{N}\) denotes the set of natural numbers.
**Equal Sets:**
Two sets are equal if they have exactly the same elements.
**Equivalent Sets:**
Two sets are equivalent if they have the same number of elements, regardless of what those elements are.
#### Given:
- \( X = \) [prime numbers less than 20]
- \( Y = \) [first six multiples of 41]
#### Set \( X \):
Prime numbers less than 20 are: 2, 3, 5, 7, 11, 13, 17, 19.
Thus,
\[ X = \{2, 3, 5, 7, 11, 13, 17, 19\} \]
Number of elements in \( X \): 8.
#### Set \( Y \):
First six multiples of 41 are: 41, 82, 123, 164, 205, 246.
Thus,
\[ Y = \{41, 82, 123, 164, 205, 246\} \]
Number of elements in \( Y \): 6.
**Comparison:**
- The sets \( X \) and \( Y \) do not contain the same elements, so they are not equal.
- The set \( X \) has 8 elements, while the set \( Y \) has 6 elements. Since they do not have the same number of elements, they are not equivalent.
**Conclusion:**
Sets \( X \) and \( Y \) are neither equal nor equivalent.
### (b) Write the Sets Using Different Methods
**Set \( X \) in the Listing Method:**
Set \( X \) is written by listing all its elements explicitly:
\[ X = \{2, 3, 5, 7, 11, 13, 17, 19\} \]
**Set \( Y \) in the Set Builder Method:**
Set \( Y \) can be described as the set of the first six multiples of 41:
\[ Y = \{41n \mid n \in \mathbb{N} \text{ and } 1 \leq n \leq 6\} \]
Here, \(\mathbb{N}\) denotes the set of natural numbers.
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