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Math Help - Powerset theory

  1. #1
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    Powerset theory

    This is kind of a simple question. Anyways, I have a set S = {1, 2, 3, 4}

    I know the powerset has 2^4 elements which is 16. I'm not sure how to find out how many of the sets in the powerset contain a certain element. I know the answer is 8 for any element.

    For example, how many of the sets in the powerset have a 1 in it. I just wrote them all out and counted and the answer is 8:
    {}, {1}, { 2}, {1, 2}, { 3}, {1, 3}, { 2, 3}, {1, 2, 3}, { 4}, {1, 4}, { 2, 4}, {1, 2, 4}, { 3, 4}, {1, 3, 4}, { 2, 3, 4}, {1, 2, 3, 4}

    ...but how do I calculate this without writing them all out?

    edit: Well I wrote up some different sets, and it always seems to be half of the number of total subsets. Is there a reason why it does this?

    Thanks.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by swtdelicaterose View Post
    This is kind of a simple question. Anyways, I have a set S = {1, 2, 3, 4}

    I know the powerset has 2^4 elements which is 16. I'm not sure how to find out how many of the sets in the powerset contain a certain element. I know the answer is 8 for any element.

    For example, how many of the sets in the powerset have a 1 in it. I just wrote them all out and counted and the answer is 8:
    {}, {1}, { 2}, {1, 2}, { 3}, {1, 3}, { 2, 3}, {1, 2, 3}, { 4}, {1, 4}, { 2, 4}, {1, 2, 4}, { 3, 4}, {1, 3, 4}, { 2, 3, 4}, {1, 2, 3, 4}

    ...but how do I calculate this without writing them all out?

    edit: Well I wrote up some different sets, and it always seems to be half of the number of total subsets. Is there a reason why it does this?

    Thanks.
    If you have elements \{1,\cdots,n\} with 2^n subsets then the number of subsets you an form without, say n, is the number of subsets you can form from \{1,\cdots,n-1\} which is 2^{n-1}.

     \frac{2^{n-1}}{2^{n}<br />
}=\frac{1}{2}
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