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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UpStudy AI · Accepted Answer

To sketch the sets $$ X $$ and $$ Y $$ and their combinations, we will follow these steps:

### Step 1: Sketch the Set $$ X $$

The set $$ X $$ is defined as:
$$
X = \left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leq 1\right\}
$$
This represents a filled circle (disk) centered at the origin $$(0, 0)$$ with a radius of 1. The boundary of the circle is included since the inequality is non-strict (≤).

### Step 2: Sketch the Set $$ Y $$

The set $$ Y $$ is defined as:
$$
Y = \left\{(x, y) \in \mathbb{R}^{2} \mid x - \frac{1}{2} \leq y \leq x + \frac{1}{2}\right\}
$$
This represents a region between two lines:
1. $$ y = x - \frac{1}{2} $$ (a line with a slope of 1, passing through $$(\frac{1}{2}, 0)$$)
2. $$ y = x + \frac{1}{2} $$ (a line with a slope of 1, passing through $$(-\frac{1}{2}, 0)$$)

The area between these two lines is shaded, and it extends infinitely in both directions along the x-axis.

### Step 3: Sketch the Combined Sets

Now we will sketch the following combinations:
1. **$$ X \cup Y $$**: The union of sets $$ X $$ and $$ Y $$.
2. **$$ X \cap Y $$**: The intersection of sets $$ X $$ and $$ Y $$.
3. **$$ X - Y $$**: The set difference, which includes points in $$ X $$ that are not in $$ Y $$.
4. **$$ Y - X $$**: The set difference, which includes points in $$ Y $$ that are not in $$ X $$.

### Step 4: Shading the Regions

1. **Shading $$ X \cup Y $$**: Shade the entire area of both the disk and the region between the two lines.
2. **Shading $$ X \cap Y $$**: Shade only the area where the disk overlaps with the region between the two lines.
3. **Shading $$ X - Y $$**: Shade the part of the disk that does not overlap with the region between the two lines.
4. **Shading $$ Y - X $$**: Shade the area between the two lines that does not overlap with the disk.

### Final Sketches

Now, let's create the sketches for each of these sets.

1. **Sketch of $$ X $$**: A filled circle centered at the origin with radius 1.
2. **Sketch of $$ Y $$**: The area between the two lines extending infinitely.
3. **Sketch of $$ X \cup Y $$**: The filled circle and the area between the lines shaded together.
4. **Sketch of $$ X \cap Y $$**: The overlapping area shaded.
5. **Sketch of $$ X - Y $$**: The part of the circle that does not overlap with the lines shaded.
6. **Sketch of $$ Y - X $$**: The area between the lines that does not overlap with the circle shaded.

These sketches will visually represent the sets and their relationships.

If you need me to generate the actual plots, please let me know!
To sketch the sets $$ X $$ and $$ Y $$ and their combinations: 1. **Set $$ X $$**: Draw a filled circle centered at the origin with a radius of 1. 2. **Set $$ Y $$**: Draw two lines $$ y = x - \frac{1}{2} $$ and $$ y = x + \frac{1}{2} $$, and shade the area between them. 3. **$$ X \cup Y $$**: Shade the entire area of both the circle and the region between the lines. 4. **$$ X \cap Y $$**: Shade the overlapping area where the circle and the lines intersect. 5. **$$ X - Y $$**: Shade the part of the circle that does not overlap with the lines. 6. **$$ Y - X $$**: Shade the area between the lines that does not overlap with the circle. These sketches will show the relationships between the sets.

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