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Thread: Unique identity

  1. #1
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    Unique identity

    Suppose that * is a binary operation on a non-empty set S. Show that if (*,S) contains an identity e then e is unique.

    Is this right?

    e1 * x = x * e1 = x

    x = e2

    e1 * e2 = e2 * e1 = e2

    then do for e2

    e2 * x = x * e2 = x

    x = e1

    e2 * e1 = e1 * e2 = e1
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  2. #2
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    Quote Originally Posted by adam_leeds View Post
    Suppose that * is a binary operation on a non-empty set S. Show that if (*,S) contains an identity e then e is unique.

    Is this right?

    e1 * x = x * e1 = x

    x = e2

    e1 * e2 = e2 * e1 = e2

    then do for e2

    e2 * x = x * e2 = x

    x = e1

    e2 * e1 = e1 * e2 = e1
    I would say "let x= e2" and "let x= e1" (so a reader won't think you are saying that a previously defined x is equal to e2) but other than that, this is fine.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    I would say "let x= e2" and "let x= e1" (so a reader won't think you are saying that a previously defined x is equal to e2) but other than that, this is fine.
    thanks, this site is brilliant.

    Ill contribute after my exams are over and done with
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  4. #4
    MHF Contributor Drexel28's Avatar
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    I think a tad shorter is to note that e=ee'=e'.
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