Let `z in C` and if `A={z:"arg"(z)=pi/4}`and `B={z:"arg"(z-3-3i)=(2pi)/3}`. Then `n(A frown B)=`

Options

A. 1
B. 2
C. 3
D. 0

Explanation

To solve the problem, we need to analyze the sets \( A \) and \( B \) defined by their arguments.

1. **Understanding Set A**:
- The set \( A \) is defined as \( A = \{ z : \arg(z) = \frac{\pi}{4} \} \).
- This means that \( A \) represents all complex numbers \( z \) that lie on the line (ray) making an angle of \( \frac{\pi}{4} \) radians (or 45 degrees) with the positive x-axis.
- In Cartesian coordinates, this line can be expressed as \( z = re^{i\frac{\pi}{4}} \) for \( r \geq 0 \), or equivalently, \( z = \frac{r}{\sqrt{2}} + i\frac{r}{\sqrt{2}} \).

2. **Understanding Set B**:
- The set \( B \) is defined as \( B = \{ z : \arg(z - 3 - 3i) = \frac{2\pi}{3} \} \).
- This means that \( B \) represents all complex numbers \( z \) such that the argument of the complex number \( z - (3 + 3i) \) is \( \frac{2\pi}{3} \).
- This can be interpreted as a line (ray) starting from the point \( (3, 3) \) and extending in the direction of the angle \( \frac{2\pi}{3} \) (which is 120 degrees).
- In Cartesian coordinates, this can be expressed as \( z = 3 + 3 + r \left( -\frac{1}{2} + i\frac{\sqrt{3}}{2} \right) \) for \( r \geq 0 \).

3. **Finding the Intersection \( A \cap B \)**:
- We need to find if there are any points \( z \) that satisfy both conditions simultaneously.
- The line representing \( A \) is \( y = x \) (since \( \tan(\frac{\pi}{4}) = 1 \)).
- The line representing \( B \) can be derived from the point \( (3, 3) \) and the slope corresponding to \( \frac{2\pi}{3} \). The slope is \( \tan(\frac{2\pi}{3}) = -\sqrt{3} \), leading to the equation \( y - 3 = -\sqrt{3}(x - 3) \).

4. **Setting the Equations Equal**:
- From \( A \): \( y = x \).
- From \( B \): \( y = -\sqrt{3}x + 3 + 3\sqrt{3} \).
- Setting these equal gives us \( x = -\sqrt{3}x + 3 + 3\sqrt{3} \).
- Rearranging yields \( x + \sqrt{3}x = 3 + 3\sqrt{3} \) or \( x(1 + \sqrt{3}) = 3 + 3\sqrt{3} \).
- Solving for \( x \) gives \( x = \frac{3 + 3\sqrt{3}}{1 + \sqrt{3}} \).

5. **Finding Corresponding y**:
- Substitute \( x \) back into \( y = x \) to find \( y \).

6. **Conclusion**:
- After solving, we find that the lines do not intersect at any point other than potentially at the point \( (3, 3) \).
- However, at \( z = 3 + 3i \), the argument of \( z - (3 + 3i) = 0 \) which does not satisfy \( \arg(z - (3 + 3i)) = \frac{2\pi}{3} \).
- Therefore, there are no points in the intersection \( A \cap B \) other than the initial point which is not included.

Thus, the number of elements in the intersection \( n(A \frown B) = 0 \).