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
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To visualize the official explanation by our mythical master, please refer the diagram below:


Only way in which we can get a sure answer is when there is no overlap.

For this case you still have an YES answer to the question for (2). The range of combined set is 5 and the sum of the ranges of T and S is 4 + 0 = 4.
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Let us take 2 sets:
1) S = {1,2,3}
2) T = {4,6}

This satisfies statement 2.
Range of S = 2
Range of T = 2
Total = 4
Range of S,T = 5

It satisfies, so far so good.

Now take t ={4,5}, with S remaining same. Now Range of S,T is 4 and not greater than the sum. Answer should be (E) I believe

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.

what if
s =( 2,3,40) &
t= (41,41,41)

In this case the range of combined set would be 41 - 2 = 39 and the sum of the ranges of S and T would be 38 + 0 = 38. So, for (2) you'll get the same YES answer to the question, we've got in the solution.

Does this make sense?
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Hi Bunuel,

Let's say:
S: 2
T: 4,6

Range of (S,T) will not be greater than individual ranges.

Let;s say:
S:1
T:7,9

Range of (S,T) will be greater than individual ranges.

Please guide

\(s_{max}\) is the greatest element of set S and \(s_{min}\) is the smallest t element of set S, so \(s_{max}>s_{min}\) (\(s_{max}-s_{min}>0\)).
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I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

What about both S and T are single element set? Let S = {0} and T = {1} . T's smallest element is larger than S's biggest element . S and T can merge into {0,1} with range to be 1. The sum of the range of S and T is 1. So (2) is insufficient. What's wrong with this logic?

The range of a single-element set is 0.
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In case S = {1, 2, 3} and T = {4, 5}.

The sum of ranges of sets S and T = 2 + 1 = 3, while the range of combined set is 4. So, you'd still have an YES answer. The correct answer to the question is B, as explained above.
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S: 2 --> range = 0
T: 4,6 --> range = 2
Combined {2, 4, 6} --> range = 4.
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Very nice question that actually tests your DS skills.For option 1, you can very easily get lost with it you just assume that you do not have the sufficient info to get to the answer. Fortunately, i was able to get it right but could have got it wrong had the second scenario in option 1 given the answer as No.

@Bunel why 0 CANNOT BE GREATER THAN Smax- Smin??

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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