1. ## Infimum of subsets

Suppose that $\displaystyle A$ and $\displaystyle B$ are non-empty subsets of $\displaystyle \mathbb{R}^2.$ Define the distance $\displaystyle d(A,B)$ between $\displaystyle A$ and $\displaystyle B$ by the formula:

$\displaystyle d(A,B)=inf\{|\mathbf{a}-\mathbf{b}|:\mathbf{a}\in A, \mathbf{b}\in B\}$

1) Explain why this infimum always exists

So because,

$\displaystyle |a-b|\geq 0 \Rightarrow$ there must be a greatest lower bound

2) Suppose that $\displaystyle A=\{(x,y) : x^2+y^2<1\} and B=\{(x,y) : x^2-y^2>9\}$. Find $\displaystyle d(A,B)$

Just from drawing a picture, I see that the smallest distance between the points on the graph is 2.

$\displaystyle |a-b|=|x^2+y^2-(x^2-y^2)|$

$\displaystyle =|2y^2|$

Now, I'll try to plug in different values of y, and so the infimum of this is $\displaystyle y=0$

When $\displaystyle y=0\Rightarrow x^2<1, x^2>9$

$\displaystyle x<1, x>3$

$\displaystyle inf\{|a-b\}=inf\{|1+0-3-0|\}=2$

3) Find 2 disjoint sets $\displaystyle A$ and $\displaystyle B$ such that $\displaystyle d(A,B)=0$

Is there any sets at all?

Not too sure if I did it correct?

Yeah I'm just edited my post to show what I've done

2. Originally Posted by acevipa
Suppose that $\displaystyle A$ and $\displaystyle B$ are non-empty subsets of $\displaystyle \mathbb{R}^2.$ Define the distance $\displaystyle d(A,B)$ between $\displaystyle A$ and $\displaystyle B$ by the formula:

$\displaystyle d(A,B)=inf\{|\mathbf{a}-\mathbf{b}|:\mathbf{a}\in A, \mathbf{b}\in B\}$

1) Explain why this infimum always exists

2) Suppose that $\displaystyle A=\{(x,y) : x^2+y^2<1\} and B=\{(x,y) : x^2-y^2>9\}$. Find $\displaystyle d(A,B)$

3) Find 2 disjoint sets $\displaystyle A$ and $\displaystyle B$ such that $\displaystyle d(A,B)=0$
I see that you have over two-hundred postings.
By now you should understand that this is not a homework service
So you need to either post some of your work on a problem or you need to explain what you do not understand about the question.

Here is a hint on #1. $\displaystyle |a-b|\ge 0$.

For #3. Consider $\displaystyle A=\{(x,y) : x^2+y^2<1\} \text{ and } B=\{(x,y) : x^2-y^2>1\}$