(c) $$ \mathcal{P}(A) \cap \mathcal{P}(B) $$ (d) $$ \mathcal{P}(A \cap B) $$ 4. Sketch the sets $$ X=\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leq 1\right\} $$ and $$ Y=\left\{(x, y) \in \mathbb{R}^{2} \left\lvert\, x-\frac{1}{2} \leq y \leq x+\frac{1}{2}\right.\right\} $$ on $$ \mathbb{R}^{2} $$. On separate drawings, shade the sets $$ X \cup Y, X \cap Y, X-Y $$, and $$ Y-X $$.

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### (c) and (d)

To find the intersection of the two power sets, consider any set $$ C $$ such that
$$
C \in \mathcal{P}(A) \quad \text{and} \quad C \in \mathcal{P}(B).
$$
This means that
$$
C \subseteq A \quad \text{and} \quad C \subseteq B.
$$
Thus, any such $$ C $$ is a subset of both $$ A $$ and $$ B $$, which is equivalent to:
$$
C \subseteq A \cap B.
$$
In other words, every element of $$\mathcal{P}(A) \cap \mathcal{P}(B)$$ is a subset of $$A \cap B$$, so
$$
\mathcal{P}(A) \cap \mathcal{P}(B) = \{\, C \mid C \subseteq A \cap B \,\} = \mathcal{P}(A \cap B).
$$
Thus, we have shown that
$$
\mathcal{P}(A) \cap \mathcal{P}(B) = \mathcal{P}(A \cap B).
$$

---

### 4. Sketching the Sets

#### Definition of the Sets

- **Set $$ X $$**:  
  $$
  X = \{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 \le 1\}
  $$
  This is the closed unit disk (a circle of radius 1 centered at the origin, including its interior).

- **Set $$ Y $$**:  
  $$
  Y = \left\{(x,y) \in \mathbb{R}^2 \;\middle|\; x-\frac{1}{2} \le y \le x+\frac{1}{2}\right\}
  $$
  This is the region between the two lines:
  $$
  y = x-\frac{1}{2} \quad \text{and} \quad y = x+\frac{1}{2}.
  $$
  The region $$ Y $$ is an infinite strip (with slope 1) of constant vertical “thickness” $$1$$.

#### Steps for Sketching

1. **Sketch $$ X $$ (the Unit Disk):**
   - Draw a circle of radius 1 centered at the origin.
   - Shade the inside of the circle to indicate that all boundary and interior points are included.

2. **Sketch $$ Y $$ (the Strip):**
   - Draw the line $$ y = x - \frac{1}{2} $$. This line passes through points such as $$(0,-\frac{1}{2})$$ and $$(\frac{1}{2}, 0)$$.
   - Draw the line $$ y = x + \frac{1}{2} $$. This line passes through points such as $$(0,\frac{1}{2})$$ and $$(-\frac{1}{2}, 0)$$.
   - Shade the region between these two lines. This region extends infinitely but is bounded by these parallel straight lines.

3. **Sketch $$ X \cup Y $$ (Union):**
   - Begin with the sketches of both $$ X $$ and $$ Y $$.
   - Shade all points that belong either to the disk $$ X $$ or to the strip $$ Y $$ (or to both). Essentially, every point that lies in either region should be shaded.

4. **Sketch $$ X \cap Y $$ (Intersection):**
   - Start with the disk $$ X $$ and the strip $$ Y $$.
   - Identify the region that is common to both. That is, shade only the portion of the disk $$ X $$ that also lies inside the strip $$ Y $$.

5. **Sketch $$ X - Y $$ (Points in $$ X $$ but not in $$ Y $$):**
   - Begin with the disk $$ X $$ as the base.
   - Remove (do not shade) the part that lies in $$ Y $$. Thus, shade only the parts of the disk that fall outside the strip.

6. **Sketch $$ Y - X $$ (Points in $$ Y $$ but not in $$ X $$):**
   - Begin with the strip $$ Y $$ as the base.
   - Remove (do not shade) the points that fall inside the disk $$ X $$. Shade only those regions of the infinite strip that lie outside the circle.

#### Tips for Drawing

- **For $$ X $$**: Clearly mark the boundary circle and use a distinct color or pattern for shading.
- **For $$ Y $$**: Draw the two boundary lines with a separate style (such as dashed lines) if needed and shade the entire band.
- **For Combination Areas**:
  - Use overlapping shading patterns or different colors on different drawings to indicate which portions belong to the union, intersection, or set differences.
  - Provide separate diagrams for each of $$ X \cup Y $$, $$ X \cap Y $$, $$ X - Y $$, and $$ Y - X $$ as requested.

These steps will guide you to both understand the set operations and enable you to produce the required sketches on $$ \mathbb{R}^2 $$.
To solve the problem, follow these steps: 1. **Understanding the Sets:** - **Set $$ X $$:** A closed unit disk centered at the origin with radius 1. - **Set $$ Y $$:** An infinite strip between the lines $$ y = x - \frac{1}{2} $$ and $$ y = x + \frac{1}{2} $$. 2. **Sketching the Sets:** - **Set $$ X $$:** Draw a circle of radius 1 centered at the origin and shade its interior. - **Set $$ Y $$:** Draw the two boundary lines and shade the region between them. 3. **Combining the Sets:** - **Union ($$ X \cup Y $$):** Shade all points in either $$ X $$ or $$ Y $$ or both. - **Intersection ($$ X \cap Y $$):** Shade the overlapping region where both $$ X $$ and $$ Y $$ overlap. - **Difference ($$ X - Y $$):** Shade the part of $$ X $$ that does not overlap with $$ Y $$. - **Difference ($$ Y - X $$):** Shade the part of $$ Y $$ that does not overlap with $$ X $$. 4. **Final Drawings:** - Create separate sketches for each of the four combinations: $$ X \cup Y $$, $$ X \cap Y $$, $$ X - Y $$, and $$ Y - X $$. By following these steps, you can accurately represent the relationships between the sets $$ X $$ and $$ Y $$ on the plane $$ \mathbb{R}^2 $$.

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