[SPLIT] Let S be a set of 7 pos. int's, max being at most 24

[wessleym]

Let S be a set of seven positive integers, the maximum of which is at most 24. Prove that the sums of the elements in all the nonempty subsets of S cannot be distinct.

My answer:

S' [proper subset] S such that S' = {[empty set],x} where x [exists in] S and S'. S' is therefore nonempty, and [empty set] + x is undefined/nondistinct.

I think my answer is correct, but I have no way to know for sure. Thank you!

 

[pka]

Not sure that I follow the question. But is seems to be a pigeonhole problem.
There are 127 non-empty subsets of the set.
Now if the maximum value were 21 and not 24 it can be done.
You should see if you have posted the exact question.
What is the exact wording?

 

[wessleym]

Actually, that is the exact wording. And 127 nonempty sets? How did you figure that out?

 

[pka]

I don’t think it can be done if the maximum value in the set is 24.
I can do it for the maximum value of 21.
A seven element set has \(\displaystyle 2^7\) subsets so it has 127 non-empty subsets.
The max sum comes from {15,16,17,18,19,20,21} or 126.
The minimum sum comes from {1} or is 1.
Thus there are 127 pigeons to go into 126 holes. So two set-sums are the same.
But it does not work that way for 24.

 

[daon]

I couldn't find a definition of ZxZ anywhere on the Internet or in my textbook.

If you are given the following definition of a function F as F: ZXZ -> Z, it says take two elements of Z and do 'something' to get another element of Z.

Note, Z-set of integers. R-set of real numbers. Q-set of rational numbers.
So, for example, if I give you a function A as A: ZXR -> Q. I am saying A is a function that takes an integer and a real and does 'something' to get a rational number. It does not identify the "rule" the function must follow, it just gives you the domain (a set you take from) and the codomain (a set which the function sends to).