Answer :
Let's go through each part of the question step by step:
Given Values:
- [tex]\( n(A) = 40 \)[/tex]
- [tex]\( n(B) = 60 \)[/tex]
- [tex]\( n(A \cup B) = 80 \)[/tex]
- [tex]\( n(A \cap B) = 20 \)[/tex] (Note: [tex]\( n(A \cup B) = 20 \)[/tex] doesn't make sense because [tex]\( n(A \cup B) \)[/tex] should be greater than or equal to both [tex]\( n(A) \)[/tex] and [tex]\( n(B) \)[/tex]. We correct it to [tex]\( n(A \cap B) \)[/tex]).
### i. Find [tex]\( n(U) \)[/tex]
We need to find the total number of elements in the universal set [tex]\( U \)[/tex].
Using the principle of inclusion and exclusion, we know:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Given values:
- [tex]\( n(A \cup B) = 80 \)[/tex]
- [tex]\( n(A) = 40 \)[/tex]
- [tex]\( n(B) = 60 \)[/tex]
- [tex]\( n(A \cap B) = 20 \)[/tex]
So,
[tex]\[ 80 = 40 + 60 - 20 \][/tex]
The calculation matches, confirming [tex]\( n(A \cup B) = 80 \)[/tex].
In this context, since [tex]\( n(A \cup B) \)[/tex] is the number of elements in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], covered by the universal set [tex]\( U \)[/tex], we can say:
[tex]\[ n(U) = 80 \][/tex]
### ii. Find the value of [tex]\( n(AB) \)[/tex]
Typically, [tex]\( n(AB) \)[/tex] might be interpreted as [tex]\( n(A \cap B) \)[/tex], which represents the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
Given:
[tex]\[ n(A \cap B) = 20 \][/tex]
So, [tex]\( n(AB) = n(A \cap B) = 20 \)[/tex].
### iii. Find [tex]\( no(A) \)[/tex]
The term [tex]\( no(A) \)[/tex] is somewhat ambiguous but might refer to the number of elements outside set [tex]\( A \)[/tex].
The total elements in [tex]\( U \)[/tex] are [tex]\( n(U) = 80 \)[/tex], and the elements in [tex]\( A \)[/tex] are [tex]\( n(A) = 40 \)[/tex].
So, elements outside [tex]\( A \)[/tex] (denoted [tex]\( no(A) \)[/tex] here) are:
[tex]\[ no(A) = n(U) - n(A) \][/tex]
[tex]\[ no(A) = 80 - 40 \][/tex]
[tex]\[ no(A) = 40 \][/tex]
### iv. Draw a Venn-Diagram of the above information
To draw the Venn-Diagram:
1. Draw two overlapping circles representing sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] inside a rectangle representing the universal set [tex]\( U \)[/tex].
2. Label the left circle as [tex]\( A \)[/tex] and the right circle as [tex]\( B \)[/tex].
3. Mark the intersection (overlapping part) of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] with 20 (since [tex]\( n(A \cap B) = 20 \)[/tex]).
4. Place the difference [tex]\( n(A) - n(A \cap B) \)[/tex] into the part of [tex]\( A \)[/tex] not overlapping [tex]\( B \)[/tex]: [tex]\( 40 - 20 = 20 \)[/tex].
5. Place the difference [tex]\( n(B) - n(A \cap B) \)[/tex] into the part of [tex]\( B \)[/tex] not overlapping [tex]\( A \)[/tex]: [tex]\( 60 - 20 = 40 \)[/tex].
Thus, the regions in the Venn Diagram:
- Left part of [tex]\( A \)[/tex]: 20
- Intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]: 20
- Right part of [tex]\( B \)[/tex]: 40
This illustrates that when considering the universal set [tex]\( U \)[/tex]:
- Total elements in [tex]\( U \)[/tex]: 80
- [tex]\( A \)[/tex] contains 40 (20 in the intersection, 20 outside it)
- [tex]\( B \)[/tex] contains 60 (20 in the intersection, 40 outside it)
- The union, [tex]\( A \cup B \)[/tex], contains 80 elements
The Venn Diagram will look like this:
```
+--------------------+
| U |
| |
| +------+ |
| A | 20 | B |
| +--+---+-----+ |
| 20 |20 | 40|
| +--+---+-----+ |
+--------------------+
```
Each number correctly sums up to 80, consistent with [tex]\( n(U) = 80 \)[/tex].
Given Values:
- [tex]\( n(A) = 40 \)[/tex]
- [tex]\( n(B) = 60 \)[/tex]
- [tex]\( n(A \cup B) = 80 \)[/tex]
- [tex]\( n(A \cap B) = 20 \)[/tex] (Note: [tex]\( n(A \cup B) = 20 \)[/tex] doesn't make sense because [tex]\( n(A \cup B) \)[/tex] should be greater than or equal to both [tex]\( n(A) \)[/tex] and [tex]\( n(B) \)[/tex]. We correct it to [tex]\( n(A \cap B) \)[/tex]).
### i. Find [tex]\( n(U) \)[/tex]
We need to find the total number of elements in the universal set [tex]\( U \)[/tex].
Using the principle of inclusion and exclusion, we know:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Given values:
- [tex]\( n(A \cup B) = 80 \)[/tex]
- [tex]\( n(A) = 40 \)[/tex]
- [tex]\( n(B) = 60 \)[/tex]
- [tex]\( n(A \cap B) = 20 \)[/tex]
So,
[tex]\[ 80 = 40 + 60 - 20 \][/tex]
The calculation matches, confirming [tex]\( n(A \cup B) = 80 \)[/tex].
In this context, since [tex]\( n(A \cup B) \)[/tex] is the number of elements in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], covered by the universal set [tex]\( U \)[/tex], we can say:
[tex]\[ n(U) = 80 \][/tex]
### ii. Find the value of [tex]\( n(AB) \)[/tex]
Typically, [tex]\( n(AB) \)[/tex] might be interpreted as [tex]\( n(A \cap B) \)[/tex], which represents the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
Given:
[tex]\[ n(A \cap B) = 20 \][/tex]
So, [tex]\( n(AB) = n(A \cap B) = 20 \)[/tex].
### iii. Find [tex]\( no(A) \)[/tex]
The term [tex]\( no(A) \)[/tex] is somewhat ambiguous but might refer to the number of elements outside set [tex]\( A \)[/tex].
The total elements in [tex]\( U \)[/tex] are [tex]\( n(U) = 80 \)[/tex], and the elements in [tex]\( A \)[/tex] are [tex]\( n(A) = 40 \)[/tex].
So, elements outside [tex]\( A \)[/tex] (denoted [tex]\( no(A) \)[/tex] here) are:
[tex]\[ no(A) = n(U) - n(A) \][/tex]
[tex]\[ no(A) = 80 - 40 \][/tex]
[tex]\[ no(A) = 40 \][/tex]
### iv. Draw a Venn-Diagram of the above information
To draw the Venn-Diagram:
1. Draw two overlapping circles representing sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] inside a rectangle representing the universal set [tex]\( U \)[/tex].
2. Label the left circle as [tex]\( A \)[/tex] and the right circle as [tex]\( B \)[/tex].
3. Mark the intersection (overlapping part) of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] with 20 (since [tex]\( n(A \cap B) = 20 \)[/tex]).
4. Place the difference [tex]\( n(A) - n(A \cap B) \)[/tex] into the part of [tex]\( A \)[/tex] not overlapping [tex]\( B \)[/tex]: [tex]\( 40 - 20 = 20 \)[/tex].
5. Place the difference [tex]\( n(B) - n(A \cap B) \)[/tex] into the part of [tex]\( B \)[/tex] not overlapping [tex]\( A \)[/tex]: [tex]\( 60 - 20 = 40 \)[/tex].
Thus, the regions in the Venn Diagram:
- Left part of [tex]\( A \)[/tex]: 20
- Intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]: 20
- Right part of [tex]\( B \)[/tex]: 40
This illustrates that when considering the universal set [tex]\( U \)[/tex]:
- Total elements in [tex]\( U \)[/tex]: 80
- [tex]\( A \)[/tex] contains 40 (20 in the intersection, 20 outside it)
- [tex]\( B \)[/tex] contains 60 (20 in the intersection, 40 outside it)
- The union, [tex]\( A \cup B \)[/tex], contains 80 elements
The Venn Diagram will look like this:
```
+--------------------+
| U |
| |
| +------+ |
| A | 20 | B |
| +--+---+-----+ |
| 20 |20 | 40|
| +--+---+-----+ |
+--------------------+
```
Each number correctly sums up to 80, consistent with [tex]\( n(U) = 80 \)[/tex].
