2.9|

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SICP Exercise 2.9

We know:

\[width = \frac{(upper bound) - (lower bound)}{2}\]

The add-interval procedure adds the lower-bound and upper-bound of both the intervals passed to it. Let the two intervals be \(x\) and \(y\):

\[width (x) = \frac{U_x - L_x}{2} \\ width (y) = \frac{U_y - L_y}{2}\]

Running:

(add-interval x y)

Let’s call this new interval sum. The width is:

\[width (sum) = \frac{U_s - L_s}{2}\]

Where:

\[U_s = U_x + U_y \\ L_s = L_x + L_y\]

Substituting:

\[\begin{align} width (sum) &= \frac{(U_x + U_y) - (L_x + L_y)}{2} \\ width (sum) &= \frac{(U_x - L_x) + (U_y - L_y)}{2} \\ width (sum) &= \frac{U_x - L_x}{2} + \frac{U_y - L_y}{2} \\ width (sum) &= width (x) + width (y) \end{align}\]

Therefore, thhe width of sum or difference of two intervals is the function pnly of the widths of the intervals being added or subtracted.