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 :
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Set : This represents the set of points in the plane where . Geometrically, this is the region above the line , excluding the points on the line itself.
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Set : This represents the set of points in the plane where . Geometrically, this is the horizontal strip between and , inclusive of the boundary lines and .
Sketching :
To visualize (the union of and ), we need to combine these two regions:
- is the region above the line , but for the union, we don't need to exclude the part where , since includes the entire strip between and .
- The result will look like the horizontal strip from to , plus all the points above the line for .
In summary, will cover:
- The entire strip from to , including all values of .
- The region above the line for points where .
Would you like me to generate a visual sketch to help illustrate this region?
Here are some relative questions you might find helpful:
- How would the region change if was defined as instead of ?
- What happens to the union if is shifted to a different range, such as ?
- How would the sketch change if we considered (the intersection) instead of the union?
- How does the slope of the line influence the shape of the region?
- Can you describe how the union changes if we modify to ?
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
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