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For addition and subtraction, the width of the result is a function of the widths of the input. For example,
[aL, aH] + [bL, bH] = [aL + bL, aH + bH].
L = Low = lower bound, H = High = upper bound
The width of this interval is
width = 1/2 * ((aH + bH) - (aL + bL)) = 1/2 * ((aH - aL) + (bH - bL)) = width of interval a + width of interval b
So, the width of the sum (or difference) of two intervals is just a function of the widths of those intervals.
For multiplication and division, the story is different. If the width of the result was a function of the widths of the inputs, then multiplying different intervals with the same widths should give the same answer. For example, multiplying a width 5 interval with a width 1 interval:
[0, 10] * [0, 2] = [0, 20] (width = 10)
The following intervals have the same widths as the corresponding ones above, but multiplying gives different results:
[-5, 5] * [-1, 1] = [-5, 5] (width = 5)
Thanks to jz for providing this solution.