Hold on...I think I might have figured it out. Just allow me time to type up my solution.
How many numbers must be selected from the set {1,2,3,4,5,6}to guarantee that at least one pair of these
numbers add up to 7?
The answer ends up involving the fact that, since there are only three subsets--namely, {1,6}, {2,5}, and {3,4}--a person is only required to select four numbers from the set. This is due to the fact that at least two of them must fall within the same subset.
I am very frustrated right now (which is probably why I can't think properly, at the moment); and I am becoming more frustrated because the book gives the impression that it is so simple. I don't understand this--not one bit. Can someone please help me?
EDIT: I am having a lot of difficulty, right now. I seem to be questioning every single thing that I do, and sometimes I can just accept an idea, even the ones that appear to be most intuitive. What's wrong with me? How did I get myself into this sort of mindset?
Let . The subsets of S, whose elements all summed together result in 7, are: , , .
Since there are only three sets, all of which contain only two elements, and , by selecting four numbers from S at least two of the elements must fall within the same subset. Although, through experimentation, it would apear that any way of selecting four numbers from S results in exactly two of the three subsets being selected.
Hello, Bashyboy!
Plato asked you the proper questions.
Think before you answer.
Let .
The pairs whose sum is 7 are: , , .
"Is it possible to select a subset of three of those that does not have that property?"
Yes, we can select one number from each of those three subsets.
. . Examples: .
"Is it possible to select a subset of four of those that does not have that property?"
No . . . selecting a fourth number will give us a sum of 7.