Let's break down the problem step-by-step and find [tex]\( n(A \cup B) \)[/tex].
### Step 1: Understand and Determine Values
Given:
- Total number of elements [tex]\( n(v) = 150 \)[/tex].
- The ratio of [tex]\( n(A) \)[/tex] to [tex]\( n(B) \)[/tex] is [tex]\( 7:9 \)[/tex].
- [tex]\( 3 \cdot n(\bar{A} \cap \bar{B}) = 2 \cdot n(A \cap B) \)[/tex].
- [tex]\( n\left( \bar{A} \cup \bar{B} \right) = 50 \)[/tex].
### Step 2: Calculate [tex]\( n(A) \)[/tex] and [tex]\( n(B) \)[/tex]
The sum of the parts of the ratio [tex]\( 7 + 9 = 16 \)[/tex].
To find [tex]\( n(A) \)[/tex] and [tex]\( n(B) \)[/tex]:
[tex]\[ n(A) = (7/16) \times 150 = 65.625 \][/tex]
[tex]\[ n(B) = (9/16) \times 150 = 84.375 \][/tex]
### Step 3: Define the Number of Intersections for Complementary Sets
Given that [tex]\( 3 \cdot n(\bar{A} \cap \bar{B}) = 2 \cdot n(A \cap B) \)[/tex], we need to find [tex]\( n(\bar{A} \cap \bar{B}) \)[/tex].
Knowing from the problem that:
[tex]\[ n(A \cap B) = \frac{3}{2} \cdot n(\bar{A} \cap \bar{B}) \][/tex]
### Step 4: Calculate [tex]\( n(\bar{A} \cap \bar{B}) \)[/tex]
From previous steps:
[tex]\[ n(\bar{A} \cap \bar{B}) = (65.625 \cdot 84.375) \cdot 2 = 11074.21875 \][/tex]
### Step 5: Apply Principle of Inclusion-Exclusion for [tex]\( n(A \cup B) \)[/tex]
Now, we'll find [tex]\( n(A \cup B) \)[/tex] using the principle of inclusion-exclusion:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(\bar{A} \cap \bar{B}) \][/tex]
[tex]\[ n(A \cup B) = 65.625 + 84.375 - 11074.21875 = -10924.21875 \][/tex]
### Step 6: Determine [tex]\( n(A \cap B) \)[/tex]
Knowing:
[tex]\[ n(A \cup B) = n(v) - n(\bar{A} \cup \bar{B}) \][/tex]
[tex]\[ n(A \cap B) = n(A) + n(B) - n(A \cup B) = 65.625 + 84.375 - (-10924.21875) \][/tex]
[tex]\[ \text{Therefore,} \; n(A \cap B) = -10924.21875 \][/tex]
### Final Result:
Thus, the number of elements in [tex]\( n(A \cup B) = -10924.21875 \)[/tex].
This detailed methodology provides how we obtained the answer as [tex]\( n(A \cup B) = -10924.21875 \)[/tex].
