M13-26

Official Solution:


Is the range of a combined set \((S, T)\) greater than the sum of the ranges of sets \(S\) and \(T\)?

The range of a set is the difference between the largest and smallest elements of the set.

\(range_t = t_{max} - t_{min}\);

\(range_s = s_{max} - s_{min}\);

Question: Is \(range_{(t \text{ and } s)} > (t_{max} - t_{min}) + (s_{max} - s_{min})\)?

(1) The largest element of \(T\) is bigger than the largest element of \(S\).

Given: \(t_{max} > s_{max}\), so the largest element of the combined set is \(t_{max}\). However, we still don't know which is the smallest element of the combined set:

If it's \(t_{min}\), then the question becomes: is \(t_{max} - t_{min} > (t_{max} - t_{min}) + (s_{max} - s_{min}\))? Or: is \(0 > s_{max} - s_{min}\)? In this case, the answer would be NO;

If it's \(s_{min}\), then the question becomes: is \(t_{max} - s_{min} > (t_{max} - t_{min}) + (s_{max} - s_{min})\)? Or: is \(t_{min} > s_{max}\)? In this case, the answer would sometimes be NO and sometimes be YES.

Not sufficient.

(2) The smallest element of \(T\) is bigger than the largest element of \(S\).

Given: \(t_{min} > s_{max}\), so the largest element of the combined set is \(t_{max}\) and the smallest element of the combined set is \(s_{min}\).

So the question becomes: is \(t_{max} - s_{min} > (t_{max} - t_{min}) + (s_{max} - s_{min})\)? Or: is \(t_{min} > s_{max}\)? This condition is given to be true, so the answer is YES. Sufficient.


Answer: B