Sticky infimum and supremum question

Question

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

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

I want to

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

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

I have that:

inf(X) = 1 and sup(X) = 3

Then:

inf(Y) = ? and sup(Y) = 3

inf(X-Y) = inf X - sup Y = 1 - 3 = -2 sup(X-Y) = sup X - inf Y = 3 - ?

I have been unable to find what inf(Y) is. I would think that inf(Y) = 1

If this is the case then sup(X-Y)= 3 - 1 = 2

I would like to know is this correct, and how would I have to go about solving part a.

I would think that I would have to represent X-Y as an interval to help solve part b.

Regards

Answer 1

a) Just follow the set construction.

$\begin{align} X-Y & = \{x-y: x\in X, y\in Y\} \\[1ex] & =\{x-y: 1\leq x\leq 3, 1< y\leq 3\} \\[1ex] & =\{x-y: -2\leq x-y< 2\} \\[2ex] \therefore\quad [1;3]-(1;3] & = [-2;2) \end{align}$

b) You should now be able to answer.