Evaluating supremum and infinum

Question

I have no verified solution for this question.

I had a few questions regarding this:

$X = [1,3]$ $Y = (1,3]$

$X-Y = \{ x-y| x\in X, y\in Y\}$

The two questions that I have are that:

$a)$ Find the value of: $X-Y$.

$b)$ Are $\sup(X-Y)$ and $\inf(X-Y)$ elements of $X-Y$?

Firstly is the answer to $a)$ Simply; $1-1=0, 3-3=0\Rightarrow X-Y= (0,0]$

And for $b)$ I have: $\sup(X)=1, \inf(X)=3$. And $\sup(Y)=3, \inf(Y)$ not possible.

$\sup(X-Y) = 1-3 = -2$

$\inf(X-Y)$ not possible.

If anyone has any feedback it would be greatly appreciated.

Answer 1

Hint: To find $X-Y$, think about what the smallest element in that set is. For that you have to take the smallest element of $X$ and subtract the largest element of $Y$ (since negating it yields the smallest element). Hence, $1-3 = -2$ is the smallest element in $X-Y$.

Answer 2

So: inf(A) = 1. sup(A)=3 inf(B) = infinity sup(B) = 3

So A-B = 3-1 = 2?

Then inf(A-B) is not possible

And

sup(A-B) = 0

Have I got this correct now?

Answer 3

The solution is [-2,2) let me know if you need further clarification.

Answer 4

The solution to part a) is [-2,2) do you understand why?