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If the mean of set S does not exceed mean of any subset of 05 Jan 2020, 20:55
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Felixvalentin
Hi Bunuel and ScottTargetTestPrep

If set S was empty, none of the conditions I, II and III must be true, yet the mean of set S would not exceed the mean of any subset of S. Hence, I chose A. I would highly appreciate if you could explain where my flaw is?

Many thanks!
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The mean and the median are concepts which make sense only for non-empty set. If you have a set with no elements, what is the mean of that set? What is the median of that set? Neither the mean nor the median exists for that set; therefore the empty set does not satisfy the requirement that "the mean of set S does not exceed the mean of any subset of set S". Sure, the question could have said "the mean of the non-empty set S ..."; but as a matter of fact, that's not really necessary because as soon as the question is talking about the mean of the set S, you know that set S cannot be empty.
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If the mean of set S does not exceed mean of any subset of 25 Jun 2020, 08:21
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If the mean of set S does not exceed mean of any subset of s , which of the following must be true about set s ?

let us consider
set s = { 2,0,15 ,100 200}

now one subset will be s1 = {0,2}

so the mean of s1 will definitely be less than mean of s.

Which is contradicted by the question .

Hence if the mean of set s is less than mean of any subset of s..

All the element in set S must be equal.

If all the elements of set S is equal..Mean and median of set s will be same ..

Hence D is the correct answer
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If the mean of set S does not exceed mean of any subset of 29 Jan 2025, 11:13
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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.
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If the mean of set S does not exceed mean of any subset of 29 Jan 2025, 11:21
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hari0616
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.
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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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If the mean of set S does not exceed mean of any subset of 18 Jul 2025, 07:41
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Can you elaborate the case B; example because as per the statement I - it has one element only ; then it could be single element set or same entities single value. Secondly the statement is mean should not be greater than then automatically mean less than and equal so i think the answer should be E all three statement are right
Bunuel
Official Solution:


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?

I. \(S\) contains only one element.

II. All elements in \(S\) are equal.

III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).


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


Consider a dataset \(S\). If the average of \(S\) is not greater than the average of ANY subset of \(S\), then the following scenarios are possible:

A. \(S=\{x\}\), where \(S\) contains only one element (e.g. {7});

B. \(S=\{x, x, ...\}\), where \(S\) contains more than one element and all elements are equal (e.g. {7,7,7,7}).

This is because if the dataset \(S\) contains two or more different elements, then we can always consider the subset with the smallest number, and the mean of this subset (mean of the subset = smallest number) will be less than the mean of the entire dataset (mean of the full dataset > smallest number).

For example, if \(S=\{3, 5\}\), then the mean of \(S=4\). If we pick the subset with the smallest number, \(s'=\{3\}\), then the mean of \(s'=3\). Thus, \(3 < 4\).

Now let's consider the statements:

I. \(S\) contains only one element. This statement is not always true since scenario B is also possible (\(S=\{x, x, ...\}\)).

II. All elements in \(S\) are equal. This statement is true for both scenarios A and B, hence always true.

III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\). This statement is true for both scenarios A and B, hence always true.

Therefore, statements II and III are always true.


Answer: D
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If the mean of set S does not exceed mean of any subset of 18 Jul 2025, 13:26
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Sunny00731
Can you elaborate the case B; example because as per the statement I - it has one element only ; then it could be single element set or same entities single value. Secondly the statement is mean should not be greater than then automatically mean less than and equal so i think the answer should be E all three statement are right
Bunuel
Official Solution:


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?

I. \(S\) contains only one element.

II. All elements in \(S\) are equal.

III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).


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


Consider a dataset \(S\). If the average of \(S\) is not greater than the average of ANY subset of \(S\), then the following scenarios are possible:

A. \(S=\{x\}\), where \(S\) contains only one element (e.g. {7});

B. \(S=\{x, x, ...\}\), where \(S\) contains more than one element and all elements are equal (e.g. {7,7,7,7}).

This is because if the dataset \(S\) contains two or more different elements, then we can always consider the subset with the smallest number, and the mean of this subset (mean of the subset = smallest number) will be less than the mean of the entire dataset (mean of the full dataset > smallest number).

For example, if \(S=\{3, 5\}\), then the mean of \(S=4\). If we pick the subset with the smallest number, \(s'=\{3\}\), then the mean of \(s'=3\). Thus, \(3 < 4\).

Now let's consider the statements:

I. \(S\) contains only one element. This statement is not always true since scenario B is also possible (\(S=\{x, x, ...\}\)).

II. All elements in \(S\) are equal. This statement is true for both scenarios A and B, hence always true.

III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\). This statement is true for both scenarios A and B, hence always true.

Therefore, statements II and III are always true.


Answer: D
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Statement I (S has only one element) is not always true. The condition says the mean of S is not greater than the mean of any subset. This is also satisfied if all elements in S are equal, such as {7, 7, 7}. So S can have more than one element.
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If the mean of set S does not exceed mean of any subset of 20 Jul 2025, 00:11
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Consider set S= {1,2,3,6,7}, and subset {7}

It satisfied the condition, while not having all elements in the set be equal

Shouldn't the answer be A?
angel2009
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?

I. \(S\) contains only one element.

II. All elements in \(S\) are equal.

III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).


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



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If the mean of set S does not exceed mean of any subset of 20 Jul 2025, 09:35
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kabirgandhi
Consider set S= {1,2,3,6,7}, and subset {7}

It satisfied the condition, while not having all elements in the set be equal

Shouldn't the answer be A?
angel2009
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?

I. \(S\) contains only one element.

II. All elements in \(S\) are equal.

III. The median of \(S\) is equal to the average (arithmetic mean) of \(S\).


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



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No, the given set does not satisfy the condition stated in the stem that "the average (arithmetic mean) of dataset S is not greater than the average (arithmetic mean) of ANY non-empty subset of S."

S cannot be {1, 2, 3, 6, 7}, because it has subsets with a smaller mean than the mean of S.
Mean of S = 19/5. Mean of {1}, which is a subset of S, is 1, so 19/5 > 1.

By the way, this doubt has been addressed in the thread. Please review it.
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If the mean of set S does not exceed mean of any subset of 23 Oct 2025, 23:11
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Then why is Option II, it is also a set of all elements with equal values. What is the difference between Option I and II as per trailing discussions.

TIA
Bunuel



Question asks: "which of the following MUST be true about set S" (not COULD be true).

I. Set S contains only one element --> it's not necessarily true as S can contain more than one element and still satisfy the requirement in stem. For example if S={3, 3, 3, 3} then mean of S equals to 3 and it does not exceed the mean of ANY subset of S, which also will be equal to 3.

Hope it's clear.
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If the mean of set S does not exceed mean of any subset of 24 Oct 2025, 00:31
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kdipayan
Then why is Option II, it is also a set of all elements with equal values. What is the difference between Option I and II as per trailing discussions.

TIA

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Both describe sets with equal elements, but statement I is more restrictive. A one-element set automatically satisfies the condition, but the question asks what must be true. The condition also holds for sets with multiple identical elements (for example {3, 3, 3}), so I is not required. Statement II covers both cases, which is why only II (and III, which follows from it) must be true.
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