1. Infimum of subsets

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

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

1) Explain why this infimum always exists

So because,

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

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

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

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

$=|2y^2|$

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

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

$x<1, x>3$

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

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

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

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

1) Explain why this infimum always exists

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

3) Find 2 disjoint sets $A$ and $B$ such that $d(A,B)=0$
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. $|a-b|\ge 0$.

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