Chewybakas said:Homework Statement
I have to sketch the following region in R^2 but i Have no idea how to do it!
Homework Equations
S1 = {(x; y) εℝ^2 : -2≤x≤4 y,≤2}
The Attempt at a Solution
I plotted the line from -2 to 4 on the x-axis and a line greater then 2 from the y-axis but i am sure i am doing this incorrectly! any help is appreciated!
Chewybakas said:So if i plot those points and draw the lines i get a rectangular shape on my graph is that the correct solution?
This is not a useful description. The region is unbounded in one direction. It is bounded on three sides by straight lines. A better description would tell us what the three boundary lines are and maybe what the points at the corners are.Chewybakas said:I have a graph with an x and y axis, the points drawn in and lines from each point, where the line y=2 intersects the x points i shade in the rectangular shape. If I am wrong could you please tell me what to do!
-2≤x≤4. Draw two vertical lines at x= -2 and at x= 4 so that the "line from -2 to 4 on the x axis" is inside them. I don't know what you mean by "a line greater than 2 from the y-axis". y≤2 is all y less than or equal to 2 so draw a horizontal line at y= 2 crossing those two vertical lines. The region you want to shade is the area below that horizontal line and between the two vertical lines. Because there is no lower limit on y, there is no lower boundary- the region you want is the "infinite strip" between the vertical lines and below the horizontal line.Chewybakas said:Homework Statement
I have to sketch the following region in R^2 but i Have no idea how to do it!
Homework Equations
S1 = {(x; y) εℝ^2 : -2≤x≤4 y,≤2}
The Attempt at a Solution
I plotted the line from -2 to 4 on the x-axis and a line greater then 2 from the y-axis but i am sure i am doing this incorrectly! any help is appreciated!
Chewybakas said:Ahh that's where i was going wrong! I was using the x-axis itself as a boundary! I have no idea why! Thank you for clearing that up! But in this question i have no figures.
{(x; y) εℝ^2 : x≤y}
Is that just the x and y axis! thank you for all your help!
HallsofIvy said:-2≤x≤4. Draw two vertical lines at x= -2 and at x= 4 so that the "line from -2 to 4 on the x axis" is inside them. I don't know what you mean by "a line greater than 2 from the y-axis". y≤2 is all y less than or equal to 2 so draw a horizontal line at y= 2 crossing those two vertical lines. The region you want to shade is the area below that horizontal line and between the two vertical lines. Because there is no lower limit on y, there is no lower boundary- the region you want is the "infinite strip" between the vertical lines and below the horizontal line.
The purpose of sketching regions in ℝ^2 is to visually represent mathematical concepts and equations. This can help with understanding and solving problems involving functions, inequalities, and geometric shapes.
The boundaries of a region in ℝ^2 can be determined by solving the equations or inequalities that define the region. This can involve finding the intersection points of lines or curves, or determining where a function changes from positive to negative or vice versa.
A closed region in ℝ^2 includes its boundary points, while an open region does not. In other words, a closed region includes the lines or curves that define it, while an open region does not include those lines or curves.
To shade a region in ℝ^2, first determine the boundaries of the region. Then, shade in the area within those boundaries. If the region is closed, include the boundary lines in the shading. If the region is open, do not include the boundary lines in the shading.
Yes, a region in ℝ^2 can be infinite in size. This can occur when the boundaries of the region are defined by equations or inequalities that do not have a finite endpoint. For example, the region defined by the inequality y > 0 would be infinite in size.