Math Problem Statement

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

Solution

We are tasked with sketching the union of two subsets of R2\mathbb{R}^2:

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

  2. Set YY: This represents the set of points (x,y)(x, y) in the plane where 0y20 \leq y \leq 2. Geometrically, this is the horizontal strip between y=0y = 0 and y=2y = 2, inclusive of the boundary lines y=0y = 0 and y=2y = 2.

Sketching XYX \cup Y:

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

  • XX is the region above the line y=xy = x, but for the union, we don't need to exclude the part where 0y20 \leq y \leq 2, since YY includes the entire strip between y=0y = 0 and y=2y = 2.
  • The result will look like the horizontal strip from y=0y = 0 to y=2y = 2, plus all the points above the line y=xy = x for y>2y > 2.

In summary, XYX \cup Y will cover:

  • The entire strip from y=0y = 0 to y=2y = 2, including all values of xx.
  • The region above the line y=xy = x for points where y>2y > 2.

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


Here are some relative questions you might find helpful:

  1. How would the region change if XX was defined as yxy \geq x instead of y>xy > x?
  2. What happens to the union XYX \cup Y if YY is shifted to a different range, such as 1y31 \leq y \leq 3?
  3. How would the sketch change if we considered XYX \cap Y (the intersection) instead of the union?
  4. How does the slope of the line y=xy = x influence the shape of the region?
  5. Can you describe how the union changes if we modify XX to y<xy < 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.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Coordinate Geometry
Union of Sets

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12