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

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

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

  • 1
    You are indeed correct, though. To solve part (a), consider what the largest and smallest elements you can obtain are by subtracting an element of Y from an element of X. – 2017-02-11

1 Answers 1

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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.

  • 0
    Great thank you a lot. Since inf(X-Y) = -2 and sup(X-Y) = 2. Then from the interval notation 2 is not bounded in the interval notation, whereas -2 is. So then inf(X-Y) is an element of X-Y and sup(X-Y) is not an element of X-Y. Is this correct. – 2017-02-12