Let X = {(x, y) ∈ R 2 : y > x} and Y = {(x, y) ∈ R 2 : 0 ≤ y ≤ 2}. Sketch the following subsets of R 2 . X ∪ Y

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.

Let X = {(x, y) ∈ R² : y > x} and Y = {(x, y) ∈ R² : 0 ≤ y ≤ 2}. Sketch the following subsets of R²: X ∪ Y.

Explore how to sketch the union of sets X = {(x, y) ∈ R² : y > x} and Y = {(x, y) ∈ R² : 0 ≤ y ≤ 2}. This problem involves key concepts from set theory and analytic geometry, demonstrating the combination of regions defined by inequalities. Suitable for high school students in Grades 10-12.

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