(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 \( Y-X \).

### (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 \( \mathbb{R}^2 \).