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Hi, why have none of the solutions considered Null set as one of the subsets of S? If you consider null set, the 2 and 3 options also break down. Bunuel
Bunuel
If the mean of set S does not exceed mean of any subset of set S, which of the following must be true about set S ?

I. Set S contains only one element
II. All elements in set S are equal
III. The median of set S equals the mean of set S


A. none of the three qualities is necessary
B. II only
C. III only
D. II and III only
E. I, II, and III


"The mean of set S does not exceed mean of any subset of set S" --> set S can be:
A. \(S=\{x\}\) - S contains only one element (eg {7});
B. \(S=\{x, x, ...\}\) - S contains more than one element and all elements are equal (eg{7,7,7,7}).

Why is that? Because if set S contains two (or more) different elements, then we can always consider the subset with smallest number and the mean of this subset (mean of subset=smallest number) will be less than mean of entire set (mean of full set>smallest number).

Example: S={3, 5} --> mean of S=4. Pick subset with smallest number s'={3} --> mean of s'=3 --> 3<4.

Now let's consider the statements:

I. Set S contains only one element - not always true, we can have scenario B too (\(S=\{x, x, ...\}\));

II. All elements in set S are equal - true for both A and B scenarios, hence always true;

III. The median of set S equals the mean of set S - - true for both A and B scenarios, hence always true.

So statements II and III are always true.

Answer: D.

non-empty was missing from the text:
If the average (arithmetic mean) of dataset \(S\) is not greater than the average (arithmetic mean) of any non-empty subset of \(S\), which of the following statements must be true?
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