Sets & Hyperplanes Homework: Convexity, Separability & More

[squenshl]

Homework Statement

Consider the sets $A = \left\{(x_1,x_2) \in\mathbb{R}^2: x_1+x_2 \leq 1\right\}$ which is a straight line going through $(0,1)$ and $(1,0)$ and $B = \left\{(x_1,x_2) \in\mathbb{R}^2: (x_1-3)^2+(x_2-3)^2 \leq 1 \right\}$ which is a circle of radius $1$ centred at $(3,3).$

1. Are the sets $A$ and $B$ convex? Are they closed? Are they compact?
2. What is $A\cap B$?
3. Is it possible to find a hyperplane that separates $A$ and $B$? That strictly separates them?
4. If they can be strictly separated then give one hyperplane that strictly separates the sets. If they can be separated but cannot be strictly separated then give one hyperplane that separates the sets and explain why they cannot be strictly separated. If they cannot be separated then explain why they cannot be separated.

Homework Equations

The Attempt at a Solution

1. I know they are convex, closed and compact by drawing a diagram. Is that sufficient enough or would I have to show it using maths? If so, how would I do that ? Some guidance would be great!
2. That is just the empty set.
3. Yes it is possible to separate them but not sure about if it strictly separates them.
4. The line $x_1+x_2=2$ is a hyperplane that strictly separates them I think. Is it possible 2 have sets that are both separated and strictly separated by a hyperplane. If not then obviously its just the first part of question 4 "If they can be strictly separated then give one hyperplane that strictly separates the sets".

Thanks a lot for the help!

 

[Daeho Ro]

Homework Statement

Consider the sets $A = \left\{(x_1,x_2) \in\mathbb{R}^2: x_1+x_2 \leq 1\right\}$ which is a straight line going through $(0,1)$ and $(1,0)$

One comment is that the set $A$ didn't mean the line, but the area below that line including itself.

 

[Mark44]

Homework Statement

Consider the sets $A = \left\{(x_1,x_2) \in\mathbb{R}^2: x_1+x_2 \leq 1\right\}$ which is a straight line going through $(0,1)$ and $(1,0)$ and $B = \left\{(x_1,x_2) \in\mathbb{R}^2: (x_1-3)^2+(x_2-3)^2 \leq 1 \right\}$ which is a circle of radius $1$ centred at $(3,3).$

Like Daeho Ro already said, set A is not just a line. Also, set B is not just the circle. It is a disk: the bounding circle and all of the points inside it.

1. Are the sets $A$ and $B$ convex? Are they closed? Are they compact?
2. What is $A\cap B$?
3. Is it possible to find a hyperplane that separates $A$ and $B$? That strictly separates them?

Hyperplane? Your sets are in $\mathbb{R}^2$. I've never heard of people referring to hyperplanes in spaces with a dimension less than 4. A "hyperplane" in $\mathbb{R}^2$ is just a line. Same comment in the next question.

4. If they can be strictly separated then give one hyperplane that strictly separates the sets. If they can be separated but cannot be strictly separated then give one hyperplane that separates the sets and explain why they cannot be strictly separated. If they cannot be separated then explain why they cannot be separated.

Homework Equations

The Attempt at a Solution

1. I know they are convex, closed and compact by drawing a diagram. Is that sufficient enough or would I have to show it using maths? If so, how would I do that ? Some guidance would be great!
2. That is just the empty set.
3. Yes it is possible to separate them but not sure about if it strictly separates them.
4. The line $x_1+x_2=2$ is a hyperplane that strictly separates them I think. Is it possible 2 have sets that are both separated and strictly separated by a hyperplane. If not then obviously its just the first part of question 4 "If they can be strictly separated then give one hyperplane that strictly separates the sets".

Thanks a lot for the help!

 

[squenshl]

Cool thanks!

So they can only be strictly separated and not just separated or is it the other way around??

 

[Daeho Ro]

This attachment will be helpful to you.

 

[squenshl]

Great thanks. A & B can be separated but not strictly separated.

 

[squenshl]

Oops $A$ & $B$ can be strictly separated by $x_1+x_2=2$