Sure, let's go through this step-by-step.
### a. Define the set
Let [tex]\( A \)[/tex] be the set with [tex]\( n(A) = 40 \)[/tex] elements.
Let [tex]\( B \)[/tex] be the set with [tex]\( n(B) = 60 \)[/tex] elements.
* The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted by [tex]\( A \cup B \)[/tex], contains [tex]\( n(A \cup B) = 80 \)[/tex] elements.
### b. Find the value of [tex]\( n(A \cap B) \)[/tex]
To find the number of elements in the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted by [tex]\( n(A \cap B) \)[/tex], we use the formula for the union of two sets:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Plugging the given values into the formula, we get:
[tex]\[ 80 = 40 + 60 - n(A \cap B) \][/tex]
Simplifying the equation:
[tex]\[ 80 = 100 - n(A \cap B) \][/tex]
Solving for [tex]\( n(A \cap B) \)[/tex]:
[tex]\[ n(A \cap B) = 100 - 80 \][/tex]
[tex]\[ n(A \cap B) = 20 \][/tex]
Thus, [tex]\( n(A \cap B) = 20 \)[/tex].
### c. Find the value of [tex]\( n(A) - n(A \cap B) \)[/tex]
This is the number of elements only in set [tex]\( A \)[/tex], i.e., the elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]:
[tex]\[ n(\text{only } A) = n(A) - n(A \cap B) \][/tex]
Given [tex]\( n(A) = 40 \)[/tex] and [tex]\( n(A \cap B) = 20 \)[/tex]:
[tex]\[ n(\text{only } A) = 40 - 20 \][/tex]
[tex]\[ n(\text{only } A) = 20 \][/tex]
Thus, the number of elements only in [tex]\( A \)[/tex] is [tex]\( 20 \)[/tex].
### d. Represent the above information in a Venn diagram
To represent the given information in a Venn diagram:
1. Draw two intersecting circles, one representing set [tex]\( A \)[/tex] and the other representing set [tex]\( B \)[/tex].
2. The intersection (common area) of these circles represents [tex]\( A \cap B \)[/tex] and contains [tex]\( 20 \)[/tex] elements.
3. The part of circle [tex]\( A \)[/tex] excluding the intersection represents the elements only in [tex]\( A \)[/tex] and contains [tex]\( 20 \)[/tex] elements.
4. The part of circle [tex]\( B \)[/tex] excluding the intersection is for the elements only in [tex]\( B \)[/tex]. Given [tex]\( n(B) = 60 \)[/tex] and [tex]\( n(A \cap B) = 20 \)[/tex], this part contains [tex]\( 60 - 20 = 40 \)[/tex] elements.
5. The total number of elements in [tex]\( A \cup B \)[/tex] (the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]) is consistent with the given [tex]\( 80 \)[/tex] elements.
Graphically, the Venn diagram would look something like this:
```
_______________
/ \
/ 20 \
| (__20__) 40 |
| A |
|\______________ /
\_____________ / B
```
In this Venn diagram:
- The left circle (A) contains 20 elements that are only in A.
- The intersection of both circles contains the 20 elements common to A and B.
- The right circle (B) outside the intersection area contains 40 elements that are only in B.
