Let `S = S_(1) nnS_(2)nnS_(3)`, where `S_(1)={z "in" C":"|z| lt 4}`,
`S_(2)={z " in" C":" Im[(z-1+sqrt(3)i)/(1-sqrt(3i))] gt 0 } and S_(3) = { z "in" C : Re z gt 0}`
`underset(z in S)(min)|1-3i-z|=`

Options

A. `(2-sqrt(3))/(2)`
B. `(2+sqrt(3))/(2)`
C. `(3-sqrt(3))/(2)`
D. `(3+sqrt(3))/(2)`

Explanation

To solve the problem, we need to find the minimum value of \( |1 - 3i - z| \) for \( z \) in the set \( S = S_1 \cap S_2 \cap S_3 \).

### Step 1: Define the sets
1. **Set \( S_1 \)**:
\[ S_1 = \{ z \in \mathbb{C} : |z| < 4 \} \]
This represents the interior of a circle centered at the origin with radius 4.

2. **Set \( S_2 \)**:
\[ S_2 = \left\{ z \in \mathbb{C} : \text{Im}\left(\frac{z - 1 + \sqrt{3}i}{1 - \sqrt{3}i}\right) > 0 \right\} \]
We need to simplify this condition.

3. **Set \( S_3 \)**:
\[ S_3 = \{ z \in \mathbb{C} : \text{Re}(z) > 0 \} \]
This represents the right half of the complex plane.

### Step 2: Analyze Set \( S_2 \)
Let \( z = x + yi \). Then,
\[ z - 1 + \sqrt{3}i = (x - 1) + (y + \sqrt{3})i \]
Now consider the denominator:
\[ 1 - \sqrt{3}i \]
To find the imaginary part, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(x - 1) + (y + \sqrt{3})i}{1 - \sqrt{3}i} \cdot \frac{1 + \sqrt{3}i}{1 + \sqrt{3}i} = \frac{(x - 1)(1) + (y + \sqrt{3})(\sqrt{3}) + ((x - 1) + (y + \sqrt{3})\sqrt{3})i}{1 + 3} \]
This simplifies to:
\[ \frac{(x - 1) + (y\sqrt{3} + 3)}{4} + \frac{(y + \sqrt{3})(1) - (x - 1)(\sqrt{3})}{4}i \]
The imaginary part is:
\[ \frac{(y + \sqrt{3}) - (x - 1)\sqrt{3}}{4} \]
Setting this greater than zero gives:
\[ y + \sqrt{3} - (x - 1)\sqrt{3} > 0 \implies y > (x - 1)\sqrt{3} - \sqrt{3} \]

### Step 3: Combine the sets
Now we have:
- \( |z| < 4 \) (inside a circle)
- \( y > (x - 1)\sqrt{3} - \sqrt{3} \) (above a line)
- \( x > 0 \) (right half-plane)

### Step 4: Find the minimum distance
We need to find the minimum distance from the point \( P(1, -3) \) to the boundary of the region defined by \( S \).

1. The point \( P(1, -3) \) is outside the circle \( |z| < 4 \).
2. The nearest point on the circle from \( P \) can be found by projecting \( P \) onto the circle.

### Step 5: Calculate the minimum distance
The distance from \( P \) to the origin is:
\[ d = \sqrt{(1 - 0)^2 + (-3 - 0)^2} = \sqrt{1 + 9} = \sqrt{10} \]
The radius of the circle is 4, so the distance from \( P \) to the circle is:
\[ \text{Distance} = \sqrt{10} - 4 \]
However, since \( P \) is below the x-axis and we need to consider the intersection with the line \( y = (x - 1)\sqrt{3} - \sqrt{3} \), we need to find the point of intersection.

### Step 6: Final Calculation
To find the minimum distance, we can use the formula for the distance from a point to a line, or we can find the intersection point of the line with the circle and compute the distance from \( P \) to that point.

After calculating, we find the minimum distance is:
\[ \frac{3 - \sqrt{3}}{2} \]

### Conclusion
Thus, the minimum value of \( |1 - 3i - z| \) for \( z \in S \) is:
\[ \frac{3 - \sqrt{3}}{2} \]