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Let's break down and solve the given problem step by step.

### Part (a) \( |z - i| = 2 \):
This is the equation for a circle in the complex plane. In general, \( |z - a| = r \) represents a circle centered at \( a \) with radius \( r \).

- \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.
- \( |z - i| = 2 \) can be rewritten as \( |(x + iy) - i| = 2 \), or \( |x + (y - 1)i| = 2 \).
- This is a circle centered at \( (0, 1) \) with radius 2.

So, the solution set for this part is a circle with center at \( (0, 1) \) and radius 2.

### Part (b) \( |z - 2 + i| < 1 \):
This is the inequality representing the interior of a circle.

- \( |z - (2 - i)| < 1 \), which means \( z \) lies inside the circle with center at \( (2, -1) \) and radius 1.

Thus, the solution set for this part is the interior of a circle with center at \( (2, -1) \) and radius 1.

### For parts (c), (d), and (e):
These are asking about the regions described by certain complex inequalities.

- **(c)** The upper half-plane is given by \( y \geq 0 \). In the complex plane, this corresponds to points where the imaginary part of \( z \) is non-negative, i.e., \( \text{Im}(z) \geq 0 \).
  
- **(d)** The left half-plane is given by \( x \leq 0 \), meaning the real part of \( z \) is non-positive, i.e., \( \text{Re}(z) \leq 0 \).

- **(e)** The exterior of the disk centered at the origin with radius 3 includes all points satisfying \( |z| > 3 \).

### Summary of answers:
1. (a) A circle centered at \( (0, 1) \) with radius 2.
2. (b) The interior of a circle centered at \( (2, -1) \) with radius 1.
3. (c) The upper half-plane.
4. (d) The left half-plane.
5. (e) The exterior of a disk with center at the origin and radius 3.

Would you like further details on how these results were derived?

Here are 5 related questions to expand on this:
1. What is the general form of a circle in the complex plane?
2. How do inequalities affect the graphical representation of regions in the complex plane?
3. Can you describe the geometric meaning of \( |z - a| = r \) for any complex number \( a \)?
4. How would the equation \( |z - i| = 2 \) change if the right-hand side were greater or less than 2?
5. What transformations occur when shifting the center of a circle in the complex plane?

**Tip:** For inequalities in the complex plane, the region inside the circle is described by \( |z - a| < r \), while the region outside the circle is \( |z - a| > r \).

Hahmottele kompleksitasoon kaikki pisteet, jotka toteuttavat yhtälön (a) |z - i| = 2, (b) |z - 2 + i| < 1. Millainen kompleksilukuepäyhtälö kuvaa kompleksitasoon pistejoukkoa, joka vastaa (c) ylempää puolitason (y >= 0), (d) vasemmanpuoleista puolitason (x <= 0), (e) origokeskisen 3-säteisen kiekon ulkopuolelle jääviä pisteitä?

This problem covers how to represent circles and inequalities in the complex plane. It involves sketching points that satisfy the conditions |z - i| = 2, |z - 2 + i| < 1, and understanding the regions in the complex plane for upper half-planes, left half-planes, and points outside a disk centered at the origin with radius 3.

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