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