Exercise 2.9: The
width of an interval
is half of the difference between its upper and lower bounds. The width is a
measure of the uncertainty of the number specified by the interval. For some
arithmetic operations the width of the result of combining two intervals is a
function only of the widths of the argument intervals, whereas for others the
width of the combination is not a function of the widths of the argument
intervals. Show that the width of the sum (or difference) of two intervals is
a function only of the widths of the intervals being added (or subtracted).
Give examples to show that this is not true for multiplication or division.
练习 2.9：区间的宽度 (width) 是其上界与下界之差的一半。宽度是对区间所表示数值不确定性的一种度量。对于某些算术操作，两个区间组合结果的宽度仅是各参数区间宽度的函数；而对于另一些操作，组合结果的宽度并不是参数宽度的函数。证明两个区间之和（或差）的宽度仅是各被加（或被减）区间宽度的函数。并举例说明这对乘法或除法不成立。