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Sagot :
Sure, let's break down the problem step-by-step:
### Understanding the given sets:
1. Set L: This set contains the elements [tex]\( \{2, 3, 5, 7\} \)[/tex].
2. Set M: This set contains the factors of 6. The factors of 6 are [tex]\( \{1, 2, 3, 6\} \)[/tex].
3. Set N: This set is given as [tex]\( \{1, 2, 3, \ldots\} \)[/tex], which represents the set of all positive integers. We interpret this to mean that set N includes all positive integers, running infinitely.
### Solution:
(a) List the elements of set [tex]\( M \)[/tex].
To determine the elements in set [tex]\( M \)[/tex], we need to list the factors of 6. The factors of 6 are the numbers that divide 6 without leaving a remainder. They are:
[tex]\[1, 2, 3, \text{and} 6.\][/tex]
Therefore, the elements of set [tex]\( M \)[/tex] are:
[tex]\[ \{ 1, 2, 3, 6 \}. \][/tex]
(b) Which of the given sets is an infinite set?
From the description:
- Set L has a finite number of elements [tex]\( \{2, 3, 5, 7\} \)[/tex].
- Set M has a finite number of elements [tex]\( \{1, 2, 3, 6\} \)[/tex].
- Set N is [tex]\( \{1, 2, 3, \ldots\} \)[/tex], which denotes the set of all positive integers and hence continues indefinitely.
Since set N includes all positive integers and does not stop, it is an infinite set.
Therefore, set N is an infinite set.
(c) State with reason whether the sets [tex]\( L \)[/tex] and [tex]\( M \)[/tex] are equal or equivalent sets.
To determine if sets [tex]\( L \)[/tex] and [tex]\( M \)[/tex] are equal:
- Two sets are equal if they contain exactly the same elements.
- Set L is [tex]\( \{2, 3, 5, 7\} \)[/tex].
- Set M is [tex]\( \{1, 2, 3, 6\} \)[/tex].
Clearly, the elements of set L and set M are not identical. Thus, sets L and M are not equal.
To determine if sets [tex]\( L \)[/tex] and [tex]\( M \)[/tex] are equivalent:
- Two sets are equivalent if they have the same number of elements, regardless of what those elements are.
- Set L has 4 elements, and set M also has 4 elements.
Since both sets L and M contain the same number of elements (4 each), sets L and M are equivalent.
### Summary of Results:
(a) The elements of set [tex]\( M \)[/tex] are:
[tex]\[ \{1, 2, 3, 6\} \][/tex]
(b) The infinite set among the given sets is:
[tex]\[ N \][/tex]
(c) Sets [tex]\( L \)[/tex] and [tex]\( M \)[/tex] are not equal but are equivalent based on the fact that they have the same number of elements.
### Understanding the given sets:
1. Set L: This set contains the elements [tex]\( \{2, 3, 5, 7\} \)[/tex].
2. Set M: This set contains the factors of 6. The factors of 6 are [tex]\( \{1, 2, 3, 6\} \)[/tex].
3. Set N: This set is given as [tex]\( \{1, 2, 3, \ldots\} \)[/tex], which represents the set of all positive integers. We interpret this to mean that set N includes all positive integers, running infinitely.
### Solution:
(a) List the elements of set [tex]\( M \)[/tex].
To determine the elements in set [tex]\( M \)[/tex], we need to list the factors of 6. The factors of 6 are the numbers that divide 6 without leaving a remainder. They are:
[tex]\[1, 2, 3, \text{and} 6.\][/tex]
Therefore, the elements of set [tex]\( M \)[/tex] are:
[tex]\[ \{ 1, 2, 3, 6 \}. \][/tex]
(b) Which of the given sets is an infinite set?
From the description:
- Set L has a finite number of elements [tex]\( \{2, 3, 5, 7\} \)[/tex].
- Set M has a finite number of elements [tex]\( \{1, 2, 3, 6\} \)[/tex].
- Set N is [tex]\( \{1, 2, 3, \ldots\} \)[/tex], which denotes the set of all positive integers and hence continues indefinitely.
Since set N includes all positive integers and does not stop, it is an infinite set.
Therefore, set N is an infinite set.
(c) State with reason whether the sets [tex]\( L \)[/tex] and [tex]\( M \)[/tex] are equal or equivalent sets.
To determine if sets [tex]\( L \)[/tex] and [tex]\( M \)[/tex] are equal:
- Two sets are equal if they contain exactly the same elements.
- Set L is [tex]\( \{2, 3, 5, 7\} \)[/tex].
- Set M is [tex]\( \{1, 2, 3, 6\} \)[/tex].
Clearly, the elements of set L and set M are not identical. Thus, sets L and M are not equal.
To determine if sets [tex]\( L \)[/tex] and [tex]\( M \)[/tex] are equivalent:
- Two sets are equivalent if they have the same number of elements, regardless of what those elements are.
- Set L has 4 elements, and set M also has 4 elements.
Since both sets L and M contain the same number of elements (4 each), sets L and M are equivalent.
### Summary of Results:
(a) The elements of set [tex]\( M \)[/tex] are:
[tex]\[ \{1, 2, 3, 6\} \][/tex]
(b) The infinite set among the given sets is:
[tex]\[ N \][/tex]
(c) Sets [tex]\( L \)[/tex] and [tex]\( M \)[/tex] are not equal but are equivalent based on the fact that they have the same number of elements.
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