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

We are tasked with sketching the union of two subsets of \( \mathbb{R}^2 \):

1. **Set \( X \):** This represents the set of points \( (x, y) \) in the plane where \( y > x \). Geometrically, this is the region above the line \( y = x \), excluding the points on the line itself.

2. **Set \( Y \):** This represents the set of points \( (x, y) \) in the plane where \( 0 \leq y \leq 2 \). Geometrically, this is the horizontal strip between \( y = 0 \) and \( y = 2 \), inclusive of the boundary lines \( y = 0 \) and \( y = 2 \).

### **Sketching \( X \cup Y \):**

To visualize \( X \cup Y \) (the union of \( X \) and \( Y \)), we need to combine these two regions:

- \( X \) is the region above the line \( y = x \), but for the union, we don't need to exclude the part where \( 0 \leq y \leq 2 \), since \( Y \) includes the entire strip between \( y = 0 \) and \( y = 2 \).
- The result will look like the horizontal strip from \( y = 0 \) to \( y = 2 \), plus all the points above the line \( y = x \) for \( y > 2 \).

In summary, \( X \cup Y \) will cover:
- The entire strip from \( y = 0 \) to \( y = 2 \), including all values of \( x \).
- The region above the line \( y = x \) for points where \( y > 2 \).

Would you like me to generate a visual sketch to help illustrate this region?

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Here are some relative questions you might find helpful:

1. How would the region change if \( X \) was defined as \( y \geq x \) instead of \( y > x \)?
2. What happens to the union \( X \cup Y \) if \( Y \) is shifted to a different range, such as \( 1 \leq y \leq 3 \)?
3. How would the sketch change if we considered \( X \cap Y \) (the intersection) instead of the union?
4. How does the slope of the line \( y = x \) influence the shape of the region?
5. Can you describe how the union changes if we modify \( X \) to \( y < x \)?

**Tip:** In general, the union of sets includes all elements in either set, while the intersection contains only the elements that both sets share.

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

Explore how to sketch the union of two subsets of R^2, where X = {(x, y) ∈ R^2 : y > x} and Y = {(x, y) ∈ R^2 : 0 ≤ y ≤ 2}. This problem focuses on set theory and coordinate geometry, suitable for high school students in Grades 10-12.

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