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Math Help - Finding The Necessary Amount Of Numbers To Select From A Set

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    Finding The Necessary Amount Of Numbers To Select From A Set

    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?
    Last edited by Bashyboy; April 23rd 2013 at 02:07 PM.
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    Re: Finding The Necessary Amount Of Numbers To Select From A Set

    Hold on...I think I might have figured it out. Just allow me time to type up my solution.
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    Re: Finding The Necessary Amount Of Numbers To Select From A Set

    Let S = \{1,2,3,4,5,6\}. The subsets of S, whose elements all summed together result in 7, are: \{1,6\}, \{2,5\}, \{3,4\}.

    Since there are only three sets, all of which contain only two elements, and \{1,6\} \cup \{2,5\} \cup \{3,4\} = S, 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.
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    Re: Finding The Necessary Amount Of Numbers To Select From A Set

    Quote Originally Posted by Bashyboy View Post
    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?

    Is it possible to select a subset of three of those that does not have that property ?

    Is it possible to select a subset of four of those that does not have that property ?
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    Re: Finding The Necessary Amount Of Numbers To Select From A Set

    Hmm, I want to say the answer to both of those questions is no.
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    Re: Finding The Necessary Amount Of Numbers To Select From A Set

    Quote Originally Posted by Bashyboy View Post
    Hmm, I want to say the answer to both of those questions is no.
    Show us a subset of three not having the property.

    Show us a subset of four not having the property.
    Last edited by Plato; April 23rd 2013 at 04:39 PM.
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    Re: Finding The Necessary Amount Of Numbers To Select From A Set

    Hello, Bashyboy!

    Plato asked you the proper questions.
    Think before you answer.


    Let S = \{1,2,3,4,5,6\}.
    The pairs whose sum is 7 are: \{1,6\}, \{2,5\}, \{3,4\}.

    "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: . \{1,2,3\},\:\{1,2,4\},\:\{1,5,3\},\:\{1,5,4\}\: \hdots


    "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.
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