Intuition for interval subtraction

Question

I'm working on a problem that involves some interval arithmetic, and while I can wrap my head around the reasoning for interval addition:

the minimum value the sum could be is the sum of the two lower bounds and the maximum value it could be is the sum of the two upper bounds

I can't yet wrap my head around the intuition behind interval subtraction, which takes the form of:

$$[x_1, x_2] - [y_1, y_2] = [x_1-y_2, x_2-y_1]$$

What is an intuitive way to explain why interval subtraction requires subtracting the upper bound, ($y_2$) from the lower bound, ($x_1$) and then the lower bound ($y_1$) from the upper bound ($x_2$)?

Answer 1

The smallest the difference $a-b$ can be is when you take $a$ to be the smallest possible already (that is $x_1$), and subtract as much as you can from it (i.e. $y_2$).

The intuition for the upper bound is the same: take $a$ as high as possible and subtract as little as you can.

Note the definition used here: $$A - B=\{x-y:(x,y)\in A\times B\}$$

Not to be mixed up with the "set minus" operation $A\setminus B$.

Answer 2

One way to get some intuition is to note (assuming that $y_1 \le y_2]$) that $-[y_1,y_2] = [-y_2,-y_1]$.

Then $[x_1,x_2]-[y_1,y_2] = [x_1,x_2] + [-y_2,-y_1] = [x_1-y_2,x_2-y_1]$.