Answer :
Certainly! Let's tackle each part of this problem step-by-step.
### Part a) List the members of the sets \( E \) and \( F \)
Set \( E \):
- \( E = \{ x : x \text{ is an even number, } x < 10 \} \)
- The even numbers less than 10 are: 0, 2, 4, 6, and 8.
- Therefore, \( E = \{0, 2, 4, 6, 8\} \).
Set \( F \):
- \( F = \{ y : y \text{ is a factor of 10} \} \)
- The factors of 10 are: 1, 2, 5, 10.
- Therefore, \( F = \{1, 2, 5, 10\} \).
### Part b) Are the sets \( E \) and \( F \) equal or equivalent? Give reason.
Equal Sets:
- Two sets are equal if they contain exactly the same elements.
- \( E = \{0, 2, 4, 6, 8\} \)
- \( F = \{1, 2, 5, 10\} \)
- Since the elements of \( E \) and \( F \) are not all the same, \( E \) and \( F \) are not equal.
Equivalent Sets:
- Two sets are equivalent if they have the same number of elements.
- \( E \) has 5 elements: 0, 2, 4, 6, 8.
- \( F \) has 4 elements: 1, 2, 5, 10.
- Since \( E \) and \( F \) do not have the same number of elements, \( E \) and \( F \) are not equivalent.
### Part c) Are the sets \( E \) and \( F \) disjoint or overlapping sets? Give reason.
Disjoint Sets:
- Two sets are disjoint if they have no elements in common.
- Common elements of \( E \) and \( F \) are elements that intersect both sets.
- \( E = \{0, 2, 4, 6, 8\} \)
- \( F = \{1, 2, 5, 10\} \)
- Both sets share the common element 2.
- Therefore, \( E \) and \( F \) are not disjoint.
Overlapping Sets:
- Two sets are overlapping if they have at least one element in common.
- Since \( E \) and \( F \) share the common element 2, they are overlapping sets.
### Part d) Show the sets \( E \) and \( F \) in a Venn-diagram
To represent the sets \( E \) and \( F \) in a Venn diagram:
1. Intersection (\( E \cap F \)):
- The common element of \( E \) and \( F \) is 2.
- Intersection = \(\{2\}\).
2. Only \( E \):
- The elements of \( E \) that are not in \( F \): 0, 4, 6, 8.
- Only \( E \) = \(\{0, 4, 6, 8\}\).
3. Only \( F \):
- The elements of \( F \) that are not in \( E \): 1, 5, 10.
- Only \( F \) = \(\{1, 5, 10\}\).
Here is the Venn diagram representation:
```
(Only E) (Intersection) (Only F)
{0, 4, 6, 8} {2} {1, 5, 10}
+------------------+------------------+
| | |
| | |
| {0, 4, 6, 8} | {2} | {1, 5, 10}
| | |
| | |
| | |
+------------------+------------------+
```
Therefore, we've broken down the problem and provided detailed answers for each part.
### Part a) List the members of the sets \( E \) and \( F \)
Set \( E \):
- \( E = \{ x : x \text{ is an even number, } x < 10 \} \)
- The even numbers less than 10 are: 0, 2, 4, 6, and 8.
- Therefore, \( E = \{0, 2, 4, 6, 8\} \).
Set \( F \):
- \( F = \{ y : y \text{ is a factor of 10} \} \)
- The factors of 10 are: 1, 2, 5, 10.
- Therefore, \( F = \{1, 2, 5, 10\} \).
### Part b) Are the sets \( E \) and \( F \) equal or equivalent? Give reason.
Equal Sets:
- Two sets are equal if they contain exactly the same elements.
- \( E = \{0, 2, 4, 6, 8\} \)
- \( F = \{1, 2, 5, 10\} \)
- Since the elements of \( E \) and \( F \) are not all the same, \( E \) and \( F \) are not equal.
Equivalent Sets:
- Two sets are equivalent if they have the same number of elements.
- \( E \) has 5 elements: 0, 2, 4, 6, 8.
- \( F \) has 4 elements: 1, 2, 5, 10.
- Since \( E \) and \( F \) do not have the same number of elements, \( E \) and \( F \) are not equivalent.
### Part c) Are the sets \( E \) and \( F \) disjoint or overlapping sets? Give reason.
Disjoint Sets:
- Two sets are disjoint if they have no elements in common.
- Common elements of \( E \) and \( F \) are elements that intersect both sets.
- \( E = \{0, 2, 4, 6, 8\} \)
- \( F = \{1, 2, 5, 10\} \)
- Both sets share the common element 2.
- Therefore, \( E \) and \( F \) are not disjoint.
Overlapping Sets:
- Two sets are overlapping if they have at least one element in common.
- Since \( E \) and \( F \) share the common element 2, they are overlapping sets.
### Part d) Show the sets \( E \) and \( F \) in a Venn-diagram
To represent the sets \( E \) and \( F \) in a Venn diagram:
1. Intersection (\( E \cap F \)):
- The common element of \( E \) and \( F \) is 2.
- Intersection = \(\{2\}\).
2. Only \( E \):
- The elements of \( E \) that are not in \( F \): 0, 4, 6, 8.
- Only \( E \) = \(\{0, 4, 6, 8\}\).
3. Only \( F \):
- The elements of \( F \) that are not in \( E \): 1, 5, 10.
- Only \( F \) = \(\{1, 5, 10\}\).
Here is the Venn diagram representation:
```
(Only E) (Intersection) (Only F)
{0, 4, 6, 8} {2} {1, 5, 10}
+------------------+------------------+
| | |
| | |
| {0, 4, 6, 8} | {2} | {1, 5, 10}
| | |
| | |
| | |
+------------------+------------------+
```
Therefore, we've broken down the problem and provided detailed answers for each part.
