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16 Sep 2014, 00:49
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Is the range of a combined set \((S, T)\) greater than the sum of the ranges of sets \(S\) and \(T\)? ?
(1) The largest element of \(T\) is bigger than the largest element of \(S\).
(2) The smallest element of \(T\) is bigger than the largest element of \(S\).
(1) The largest element of \(T\) is bigger than the largest element of \(S\).
(2) The smallest element of \(T\) is bigger than the largest element of \(S\).
16 Sep 2014, 00:50
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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
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
General Discussion
31 Aug 2016, 04:46
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I think this is a high-quality question and I agree with explanation.
23 Oct 2016, 06:19
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Bunuel
Hi Bunuel,
Wont the hypothesis break down in the following screnario
T=[1,2,3,4,5]
S=[0,0,0,0]
so the answer to the question would be "E" cannot be determined.
Kindly let me know if i have misunderstood.
