Let's solve the problem step-by-step.
### Given Data:
1. The total number of elements in set [tex]\( U \)[/tex] is [tex]\( n(U) = 50 \)[/tex].
2. The ratio of [tex]\( n(A) \)[/tex] to [tex]\( n(B) \)[/tex] is [tex]\( 7:9 \)[/tex]. So, let [tex]\( n(A) = 7x \)[/tex] and [tex]\( n(B) = 9x \)[/tex], for some [tex]\( x \)[/tex].
3. The relationship [tex]\( 3n(\overline{A} \cap \overline{B}) = 2n(A \cap B) \)[/tex].
### Definitions:
1. [tex]\( \overline{A} \)[/tex] is the complement of [tex]\( A \)[/tex].
2. [tex]\( \overline{B} \)[/tex] is the complement of [tex]\( B \)[/tex].
3. [tex]\( n(\overline{A} \cap \overline{B}) \)[/tex] represents the number of elements neither in [tex]\( A \)[/tex] nor in [tex]\( B \)[/tex].
### Understanding the Complement Relationship:
From the third point:
[tex]\[ n(\overline{A} \cap \overline{B}) = n(U) - n(A \cup B) \][/tex]
### Sum of Complements:
We know:
[tex]\[ 3n(\overline{A} \cap \overline{B}) = 2n(A \cap B) \][/tex]
### Finding [tex]\( n(A \cup B) \)[/tex]:
We also have:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
### Substituting in the Values:
Substituting [tex]\( n(A) = 7x \)[/tex] and [tex]\( n(B) = 9x \)[/tex]:
### Total Number of Elements Not in Either [tex]\( A \)[/tex] or [tex]\( B \)[/tex]:
[tex]\[ n(U) - n(A \cup B) = \overline{n(A \cap B) \text {(Derived)}} \][/tex]
Therefore, combining these equations:
[tex]\[ n(U) - (n(A \cup B)) = 2 \cdot n(A \cap B) \][/tex]
We have:
[tex]\[ 50 - n(A \cup B) = 2 \cdot n(A \cap B) \][/tex]
#### Detailed Calculation:
Using [tex]\( n(A \cup B) = 7x + 9x - n(A \cap B) = 16x - n(A \cap B) \)[/tex]:
Substitute [tex]\( n(A \cup B) \)[/tex] into the equation:
[tex]\[ 50 - (16x - n(A \cap B)) = 2n(A \cap B) \][/tex]
[tex]\[ 50 - 16x + n(A \cap B) = 2n(A \cap B) \][/tex]
[tex]\[ 50 - 16x = n(A \cap B) \][/tex]
Now, we substitute [tex]\( n(A \cap B) = 50 - 16x \)[/tex]:
[tex]\[ 3(50 - n(A \cup B)) = 2(50 - 16x) \][/tex]
[tex]\[ 3(50 - (16x - (50 - 16x))) = 2(50 - 16x) \][/tex]
[tex]\[ 3(50) = 2(50 - 16x) \][/tex]
### Solving for [tex]\( x \)[/tex]:
Finally:
[tex]\[ 3(50 - 16x) = 2n(A \cap B) \][/tex]
[tex]\[ n(A \cap B) = 50 - 16x \][/tex]
[tex]\[ 50 = 2 \cdot 50 \Rightarrow 2x = \frac{3}{16} \cdot 25 \][/tex]
[tex]\[ \text { Therefore, } \][/tex]
### Conclusion:
* [tex]\( n(A \cup B) = 50 \)[/tex]:
1. [tex]\( n(U) = 50 \)[/tex],
2. [tex]\( x = 1.5625 \)[/tex],
3. [tex]\( n(A \cup B) = 0.0 \)[/tex]
Therefore, the final number of elements in the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
n(A \cup B) = 0.0
\][/tex]
