Nested Region Inclusions

★ 7/10Set Theory Absolute Value conic sections analytic geometry

Problem

Consider the three point sets in the plane

M=\{(x,y)\mid |x|+|y|<1\},

N=\left\{(x,y)\;\middle|\;\sqrt{\left(x-\tfrac 12\right)^2+\left(y+\tfrac 12\right)^2}+\sqrt{\left(x+\tfrac 12\right)^2+\left(y-\tfrac 12\right)^2}<2\sqrt 2\right\},

P=\{(x,y)\mid |x+y|<1,\ |x|<1,\ |y|<1\}.

Determine, with proof, the inclusion relationships among $M$ , $N$ , $P$ .

Answer

Solution

Difficulty	7/10
Topics	Set Theory, Absolute Value, conic sections, analytic geometry

Whiteboard

Your sketch is saved only in this browser. To share it, export your drawing as an image (whiteboard menu → Export as → PNG), then upload that image in the comments below.

Discussion

Ask questions, share alternate solutions, and use LaTeX freely.

0 comments

No comments yet.