# What is the meaning of this analysis problem and give some hint please?

What is the meaning of this analysis problem and give some hint please?

This problem was founded on Analysis 1 by Herbert Amann and Joachim Escher on page 100.

Determine the following subsets of $\Bbb R^2$ by drawing:

$$A = \{(x,y) \in \Bbb R^2 : |x-1| + |y+1| \leq 1\},$$

$$B = \{(x,y) \in \Bbb R^2 : 2x^2+y^2>1, |x| \leq |y|\},$$

$$C = \{(x,y) \in \Bbb R^2 : x^2-y^2>1, x-2y<1, y-2x<1\}.$$

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Draw them? The question seems incomplete. – ncmathsadist Jul 13 '12 at 1:22
@ncmathsadist - That is the whole problem in the book. – Victor Jul 13 '12 at 1:24
It could ask for descriptions, either by a picture or in words: "a square rotated by 45 degrees", "the exterior of an ellipse within two opposing quadrants", etc. – user31373 Jul 13 '12 at 1:26

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I would assume words. The point of the exercises is (probably) to make you fluent in reading and interpreting set definitions like these. – Robert Mastragostino Jul 13 '12 at 1:29

## 1 Answer

For example C, there are three regions of the plane defined. To find each one, change the inequality to an equals sign and plot the graph, then decide which of the graph is the region of interest. The first is a hyperbola opening toward the $+x$ and $-x$ axes. You want the regions that do not contain the origin. The second is a line of slope $\frac 12$ going through $(1,0)$ and you want the half-plane above it. The third is a line of slope $2$ going through $(0,1)$ and you want the half-plane below it. If you plot all three on the same axes and shade the regions of interest, your result is the area shaded all three ways, the intersection of the regions of each inequality. Since the inequalities are strict, the dividing lines are not included.

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To Ross Millikan - Oh, i think the college level question would be to ask the region of area that satisfy A,B,C... – Victor Jul 13 '12 at 1:45

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@Victor: I read it to be a three part question, with separate answers for A, B, and C. Each one has more than one region, so you are already asked to find the intersection. – Ross Millikan Jul 13 '12 at 1:55
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