Exploring C+D in R^2 with Euclidean Distance

[Metric_Space]

Homework Statement

Let X=R^2 and the distance be the usual Euclidean distance. If C and D are non-empty sets of R^2 and we have:

C+D := {y ϵ R^2 | there exists c ϵ C and dϵD s.t c+d = y}

A) What is C+D if the open balls are C= ball((0.5,0.5);2) and D=ball((0.5,2.5);1)

B) Same as A) expect D is now a closed ball

C) same as a) except D={(l,-1)|l ϵ R}

D) Is the following true? If C,D are non-empty subsets of R^2 s.t C is open, then the sum C+D is open.

Homework Equations

The Attempt at a Solution

A) It to, C+D is just the union set of all points in C and D s.t. for all c ϵ C and dϵD, c + d = x.

I think this set is open, but not sure how to describe it in more detail. Any ideas?

B) Same description as A but with boundary too?

C) not sure

D) I think this is false since C could be open and D could be closed, and so their sum would not be open.

 

[micromass]

Hi metric_space!

Homework Statement

Let X=R^2 and the distance be the usual Euclidean distance. If C and D are non-empty sets of R^2 and we have:

C+D := {y ϵ R^2 | there exists c ϵ C and dϵD s.t c+d = y}

A) What is C+D if the open balls are C= ball((0.5,0.5);2) and D=ball((0.5,2.5);1)

Let's make the problem a bit easier first by taking D just a point of the ball.
For example, (0.5,2.5) is an element of D, so (0.5,2.5)+C will be a subset of C+D. But what does (0.5,2.5) look like?
Try it for some other elements, for example (1,2.5) will be in D. So what does (1,2.5)+C look like?

In general, the set C+D will be the union of all sets d+C with d in D. Thus

$$C+D=\bigcup_{d\in D}(d+C)$$

So once you've figured out how the d+C look like, you'll know what the C+D looks like...

B) Same as A) expect D is now a closed ball

C) same as a) except D={(l,-1)|l ϵ R}

Same remarks as above.

D) Is the following true? If C,D are non-empty subsets of R^2 s.t C is open, then the sum C+D is open.

This is in fact true! Try to figure this out by using my remarks above...

 

[Metric_Space]

hm..not sure how to proceed

 

[micromass]

Well, what will (0.5,2.5)+C look like?

 

[Metric_Space]

Just that one point?

 

[micromass]

No, it will be more than one point. In fact

$$(0.5,2.5)+C=\{(0.5,2.5)+c~\vert~c\in C\}$$

try to sketch this...

 

[Metric_Space]

That's the point (0.5,2.5) plus the circle centered at (0.5,0.5) with radius 2?

 

[micromass]

No

A typical element of (0.5,2.5)+C has the form (0.5,2.5)+c with [itex]c\in C[/tex]. Maybe take some elements c in C and sketch (0.5,2.5)+c. Maybe you can see what happens then?

 

[Metric_Space]

I can't see what happens -- I have the sketch but that's not helping.

 

[micromass]

Try sketching 10-15 points, you'll see what happens soon enough!

 

[Metric_Space]

does the set consists of a half circle that's the intersection between C + D?

 

[micromass]

No. If I told you that (0.5,2.5)+C is an open ball, could you perhaps figure out which one?

 

[Metric_Space]

no..not off hand

 

[Metric_Space]

I'm just drawing a circle with a pt...but unable to make progress...I think I need a hint

 

[Slats18]

Think of it as a transformation. You've got some ball C, centered at (0.5,0.5) with a radius of 2. Now, C+(0.5,2.5) is the set of all points in C, in addition with the point (0.5,2.5). So, where in R^2 are these points now?