To sketch the sets $X = [-1, 3] \times [0, 2]$ and $Y = [2, 4] \times [2, 4]$ on a coordinate plane, let's first break down their structure:

Set $X = [-1, 3] \times [0, 2]$:

• The first interval, $[-1, 3]$, represents the x-coordinate range. So, $X$ extends from $x = -1$ to $x = 3$.
• The second interval, $[0, 2]$, represents the y-coordinate range. So, $Y$ extends from $y = 0$ to $y = 2$.

Therefore, $X$ is a rectangle with:

• A width (horizontal) of $3 - (-1) = 4$
• A height (vertical) of $2 - 0 = 2$

Set $Y = [2, 4] \times [2, 4]$:

• The first interval, $[2, 4]$, represents the x-coordinate range, so $Y$ extends from $x = 2$ to $x = 4$.
• The second interval, $[2, 4]$, represents the y-coordinate range, so $Y$ extends from $y = 2$ to $y = 4$.

Therefore, $Y$ is also a rectangle with:

• A width of $4 - 2 = 2$
• A height of $4 - 2 = 2$

Sketch Description:

• $X$ will be a rectangle starting at $(-1, 0)$ and ending at $(3, 2)$.
• $Y$ will be a smaller rectangle starting at $(2, 2)$ and ending at $(4, 4)$.

Now, I'll generate the visual sketch. matplotlib.pyplot as plt import matplotlib.patches as patches

Create the plot

fig, ax = plt.subplots()

Add the rectangle for X = [-1, 3] x [0, 2]

rect_X = patches.Rectangle((-1, 0), 4, 2, linewidth=2, edgecolor='blue', facecolor='blue', alpha=0.3, label="X = [-1, 3] x [0, 2]") ax.add_patch(rect_X)

Add the rectangle for Y = [2, 4] x [2, 4]

rect_Y = patches.Rectangle((2, 2), 2, 2, linewidth=2, edgecolor='red', facecolor='red', alpha=0.3, label="Y = [2, 4] x [2, 4]") ax.add_patch(rect_Y)

Set the x and y axis limits

ax.set_xlim(-2, 5) ax.set_ylim(-1, 5)

Set the labels and title

ax.set_xlabel('x') ax.set_ylabel('y') ax.set_title('Sketch of X and Y')

Display the legend

ax.legend()

Show the plot

plt.grid(True) plt.gca().set_aspect('equal', adjustable='box') plt.show()Here is the sketch showing the two sets:

• The blue rectangle represents $X = [-1, 3] \times [0, 2]$.
• The red rectangle represents $Y = [2, 4] \times [2, 4]$.

You can see how these rectangles overlap at the region where their x and y coordinates intersect.

Would you like further details or have any questions about this? Here are a few follow-up questions to consider:

1. What would happen if you modified the intervals for $X$ or $Y$?
2. How can we compute the intersection area of $X$ and $Y$?
3. What is the distance between the center of the two rectangles?
4. How would the sketch change if the intervals were swapped (e.g., $X = [0, 2] \times [-1, 3]$)?
5. What happens if $X$ and $Y$ don't overlap?

Tip: When comparing sets like these, it's useful to check for intersections by looking at their x and y ranges individually.