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Sketching Regions in ℝ^2

  • Thread starter Chewybakas
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  • #1

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!
 

Answers and Replies

  • #2
LCKurtz
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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!
Start by plotting the equality parts: x = -2, x=4, y = 2, three straight lines. They divide the plane up into several regions. Then, examining each region, shade the ones for which x and y both satisfy the inequalities.
 
  • #3
So if i plot those points and draw the lines i get a rectangular shape on my graph is that the correct solution?
 
  • #4
LCKurtz
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So if i plot those points and draw the lines i get a rectangular shape on my graph is that the correct solution?
Hard to say without seeing it. Describe where you have shaded.
 
  • #5
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 im wrong could you please tell me what to do!
 
  • #6
33,079
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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 im wrong could you please tell me what to do!
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.
 
  • #7
HallsofIvy
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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!
-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.
 
  • #8
Ahh thats 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!
 
  • #9
33,079
4,782
Ahh thats 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!
No. Start by graphing the line x = y. This line divides the plane into two regions. The region you want is one of them, including the boundary line.

To determine which half plane is the right one, pick a point not on the line and see if it satisifies x ≤ y.
 
  • #10
Ah y=x will give me a 45° line through the origin?
 
  • #11
LCKurtz
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-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.
Doesn't that count as "working the problem for them"?
 
  • #12
I was in desperate need of help, I was obviously getting it horribly wrong. He just helped me out greatly and i cant thank him enough for it!
 
  • #13
HallsofIvy
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No, I agree with the others. I should have stopped at
"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."

And perhaps that is a little too much. Just noting that the lines are vertical and horizontal should have been enough.
 

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