Let's analyze and sketch the sets $X$ and $Y$ in $\mathbb{R}^2$, and then find their union $X \cup Y$.

1. Set $X = \{(x, y) \in \mathbb{R}^2 : y > x\}$

This is the region where the $y$-coordinate is greater than the $x$-coordinate. It represents the half-plane above the line $y = x$ in $\mathbb{R}^2$. The line $y = x$ is not included in this region, so it's a strict inequality.

2. Set $Y = \{(x, y) \in \mathbb{R}^2 : 0 \leq y \leq 2\}$

This is the region where the $y$-coordinate lies between 0 and 2, inclusive of both boundaries. In other words, it's a horizontal strip in the plane bounded by the lines $y = 0$ and $y = 2$, with these boundaries included.

3. Union of $X \cup Y$

The union of the two sets, $X \cup Y$, consists of all points that are either in $X$ or in $Y$, or in both. This will be the area:

• Above the line $y = x$ (for the region $X$).
• Between $y = 0$ and $y = 2$ (for the region $Y$).

Since $X \cup Y$ includes both regions, we get a region that consists of all points above the line $y = x$, and everything between $y = 0$ and $y = 2$.

Sketch Explanation

• The line $y = x$ acts as a boundary for the region $X$, but only points above this line belong to $X$.
• The strip between $y = 0$ and $y = 2$ covers all points in that horizontal range, regardless of $x$-values.
• In the final sketch of $X \cup Y$, we will include all points from both regions.

Would you like me to generate a detailed sketch for this?

5 Related Questions:

1. How do we determine if a point belongs to a union of two sets?
2. How does the strict inequality in $y > x$ affect the region's boundary?
3. What would the intersection $X \cap Y$ look like?
4. How would the sketch change if $X$ had a different boundary, such as $y \geq x$?
5. How can this concept extend to three-dimensional spaces?

Tip: When dealing with inequalities in 2D space, think of the line as a boundary and focus on the side the inequality points to.