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Sagot :
### Step-by-Step Solution:
We are given two sets, \(E\) and \(F\), defined as follows:
1. \( E = \{ x : x \text{ is an even number}, x < 10 \} \)
2. \( F = \{ y : y \text{ is a factor of 10} \} \)
Let's go through each part of the question:
#### a) List the members of the sets \(E\) and \(F\):
- Set \( E \):
- The even numbers less than 10 are 0, 2, 4, 6, and 8.
- Therefore, \( E = \{ 0, 2, 4, 6, 8 \} \).
- Set \( F \):
- The factors of 10 are the numbers that can exactly divide 10 without leaving a remainder. These are 1, 2, 5, and 10.
- Therefore, \( F = \{ 1, 2, 5, 10 \} \).
So, the members of the sets are:
- \( E = \{ 0, 2, 4, 6, 8 \} \)
- \( F = \{ 1, 2, 5, 10 \} \)
#### b) Are the sets \(E\) and \(F\) equal or equivalent? Give reason.
- Two sets are equal if they have precisely the same elements. Comparing \( E \) and \( F \):
- \( E = \{ 0, 2, 4, 6, 8 \} \)
- \( F = \{ 1, 2, 5, 10 \} \)
- These sets do not have the same elements, so they are not equal.
- Two sets are equivalent if they have the same number of elements (cardinality). Comparing the number of elements:
- \( E \) has 5 elements.
- \( F \) has 4 elements.
- Since the number of elements is different, they are not equivalent.
Therefore, the sets \(E\) and \(F\) are neither equal nor equivalent.
#### c) Are the sets \(E\) and \(F\) disjoint or overlapping? Give reason.
- Two sets are disjoint if they have no elements in common.
- Two sets are overlapping if they have at least one element in common.
Comparing the elements:
- The common element between \( E \) and \( F \) is 2, as it appears in both sets:
- \( E = \{ 0, 2, 4, 6, 8 \} \)
- \( F = \{ 1, 2, 5, 10 \} \)
Since there is an element (2) that is common to both sets, the sets are overlapping.
#### d) Show the sets \(E\) and \(F\) in a Venn-diagram:
To visualize the relationship between \( E \) and \( F \) in a Venn diagram, we draw two overlapping circles. The overlap represents their common element(s). Here’s the Venn diagram:
```
_________
/ \
/ E \
| {0, 2, |
| 4, 6, 8} |
\ /\
\ / \
-------- \
| \ \
| | \ F |
| | {1, 2, 5, 10}
| |
----------
```
The overlapping section will contain:
- 2 (common between \( E \) and \( F \))
Thus, the Venn diagram accurately shows the relationship between the sets [tex]\(E\)[/tex] and [tex]\(F\)[/tex].
We are given two sets, \(E\) and \(F\), defined as follows:
1. \( E = \{ x : x \text{ is an even number}, x < 10 \} \)
2. \( F = \{ y : y \text{ is a factor of 10} \} \)
Let's go through each part of the question:
#### a) List the members of the sets \(E\) and \(F\):
- Set \( E \):
- The even numbers less than 10 are 0, 2, 4, 6, and 8.
- Therefore, \( E = \{ 0, 2, 4, 6, 8 \} \).
- Set \( F \):
- The factors of 10 are the numbers that can exactly divide 10 without leaving a remainder. These are 1, 2, 5, and 10.
- Therefore, \( F = \{ 1, 2, 5, 10 \} \).
So, the members of the sets are:
- \( E = \{ 0, 2, 4, 6, 8 \} \)
- \( F = \{ 1, 2, 5, 10 \} \)
#### b) Are the sets \(E\) and \(F\) equal or equivalent? Give reason.
- Two sets are equal if they have precisely the same elements. Comparing \( E \) and \( F \):
- \( E = \{ 0, 2, 4, 6, 8 \} \)
- \( F = \{ 1, 2, 5, 10 \} \)
- These sets do not have the same elements, so they are not equal.
- Two sets are equivalent if they have the same number of elements (cardinality). Comparing the number of elements:
- \( E \) has 5 elements.
- \( F \) has 4 elements.
- Since the number of elements is different, they are not equivalent.
Therefore, the sets \(E\) and \(F\) are neither equal nor equivalent.
#### c) Are the sets \(E\) and \(F\) disjoint or overlapping? Give reason.
- Two sets are disjoint if they have no elements in common.
- Two sets are overlapping if they have at least one element in common.
Comparing the elements:
- The common element between \( E \) and \( F \) is 2, as it appears in both sets:
- \( E = \{ 0, 2, 4, 6, 8 \} \)
- \( F = \{ 1, 2, 5, 10 \} \)
Since there is an element (2) that is common to both sets, the sets are overlapping.
#### d) Show the sets \(E\) and \(F\) in a Venn-diagram:
To visualize the relationship between \( E \) and \( F \) in a Venn diagram, we draw two overlapping circles. The overlap represents their common element(s). Here’s the Venn diagram:
```
_________
/ \
/ E \
| {0, 2, |
| 4, 6, 8} |
\ /\
\ / \
-------- \
| \ \
| | \ F |
| | {1, 2, 5, 10}
| |
----------
```
The overlapping section will contain:
- 2 (common between \( E \) and \( F \))
Thus, the Venn diagram accurately shows the relationship between the sets [tex]\(E\)[/tex] and [tex]\(F\)[/tex].
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