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Sagot :
Certainly! Let's go through each part of the question step-by-step.
### (i) Are the sets A and B disjoint or overlapping? Give reason.
Definition:
- Disjoint sets have no elements in common.
- Overlapping sets have at least one element in common.
Given Sets:
- [tex]\( A = \{1, 2, 3, 4, 5\} \)[/tex]
- [tex]\( B = \{5, 10, 15\} \)[/tex]
Analysis:
To determine if sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are disjoint or overlapping, we compare their elements.
- Set [tex]\( A \)[/tex] contains the elements: 1, 2, 3, 4, 5.
- Set [tex]\( B \)[/tex] contains the elements: 5, 10, 15.
Common Elements:
- Both sets include the element 5.
Since [tex]\( A \)[/tex] and [tex]\( B \)[/tex] share the element 5, they are not disjoint. Instead, they are overlapping.
Conclusion: [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are overlapping because they have at least one element in common (the element 5).
### (ii) Write the cardinal numbers of each of the sets A and B.
Definition:
- Cardinal number of a set is the number of elements in the set.
Given Sets:
- [tex]\( A = \{1, 2, 3, 4, 5\} \)[/tex]
- [tex]\( B = \{5, 10, 15\} \)[/tex]
Cardinal Number of Set A:
- Set [tex]\( A \)[/tex] has 5 elements: 1, 2, 3, 4, 5.
- Therefore, the cardinal number of [tex]\( A \)[/tex] is 5.
Cardinal Number of Set B:
- Set [tex]\( B \)[/tex] has 3 elements: 5, 10, 15.
- Therefore, the cardinal number of [tex]\( B \)[/tex] is 3.
Conclusion:
- The cardinal number of [tex]\( A \)[/tex] is 5.
- The cardinal number of [tex]\( B \)[/tex] is 3.
### (iii) Show the sets A and B in a diagram.
Since sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are overlapping, we can represent them using a Venn diagram.
```
1 2 3 4
---------- ----------
| | | |
| A | | B |
| | 5 | 10 |
---------- ----------
| 15 |
----------
```
In this Venn diagram:
- Circle [tex]\( A \)[/tex] contains the elements {1, 2, 3, 4, 5}.
- Circle [tex]\( B \)[/tex] contains the elements {5, 10, 15}.
- The intersection contains the element 5, which is common to both sets.
This visually demonstrates the overlapping nature of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
### (i) Are the sets A and B disjoint or overlapping? Give reason.
Definition:
- Disjoint sets have no elements in common.
- Overlapping sets have at least one element in common.
Given Sets:
- [tex]\( A = \{1, 2, 3, 4, 5\} \)[/tex]
- [tex]\( B = \{5, 10, 15\} \)[/tex]
Analysis:
To determine if sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are disjoint or overlapping, we compare their elements.
- Set [tex]\( A \)[/tex] contains the elements: 1, 2, 3, 4, 5.
- Set [tex]\( B \)[/tex] contains the elements: 5, 10, 15.
Common Elements:
- Both sets include the element 5.
Since [tex]\( A \)[/tex] and [tex]\( B \)[/tex] share the element 5, they are not disjoint. Instead, they are overlapping.
Conclusion: [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are overlapping because they have at least one element in common (the element 5).
### (ii) Write the cardinal numbers of each of the sets A and B.
Definition:
- Cardinal number of a set is the number of elements in the set.
Given Sets:
- [tex]\( A = \{1, 2, 3, 4, 5\} \)[/tex]
- [tex]\( B = \{5, 10, 15\} \)[/tex]
Cardinal Number of Set A:
- Set [tex]\( A \)[/tex] has 5 elements: 1, 2, 3, 4, 5.
- Therefore, the cardinal number of [tex]\( A \)[/tex] is 5.
Cardinal Number of Set B:
- Set [tex]\( B \)[/tex] has 3 elements: 5, 10, 15.
- Therefore, the cardinal number of [tex]\( B \)[/tex] is 3.
Conclusion:
- The cardinal number of [tex]\( A \)[/tex] is 5.
- The cardinal number of [tex]\( B \)[/tex] is 3.
### (iii) Show the sets A and B in a diagram.
Since sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are overlapping, we can represent them using a Venn diagram.
```
1 2 3 4
---------- ----------
| | | |
| A | | B |
| | 5 | 10 |
---------- ----------
| 15 |
----------
```
In this Venn diagram:
- Circle [tex]\( A \)[/tex] contains the elements {1, 2, 3, 4, 5}.
- Circle [tex]\( B \)[/tex] contains the elements {5, 10, 15}.
- The intersection contains the element 5, which is common to both sets.
This visually demonstrates the overlapping nature of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
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