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Does {a^(-2)a, a^(-1)} form a set?

  1. Sep 24, 2008 #1

    tgt

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    Does {a^(-2)a, a^(-1)} form a set?
     
  2. jcsd
  3. Sep 24, 2008 #2

    CRGreathouse

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    Re: sets?

    I'm not sure what you mean. If a is just a member of some arbitrary group, then [itex]a^{-2}a=a^{-1}[/itex] and so [itex]\{a^{-2}a,a^{-1}\}=\{a^{-1}\}.[/itex] But it's still a set.

    But perhaps you mean something else?
     
  4. Sep 24, 2008 #3

    tgt

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    Re: sets?

    A set cannot contain repeated elements so if a is a member of a group then
    [itex]\{a^{-2}a,a^{-1}\}[itex] wouldn't be a set?
     
  5. Sep 24, 2008 #4
    Re: sets?

    According to what axiom in axiomatic set theory you created that set????
     
  6. Sep 25, 2008 #5
    Re: sets?

    In regular notation, we would say that {x} = {x, x} = {x, x, x}, etc.

    It's not that a set can't contain duplicate entries... it's that you aren't allowed to ask "how many copies" of something are in a set.

    Here's an example of something quite similar. The image (sometimes called the range) of a real function f, is the set

    [tex]\{f(x) | x \in R\}[/tex]

    Let f be the sine function. The image is then [tex]\{sin(x) | x \in R\}[/tex].

    Notice that this means that [tex]sin(0), sin(\pi), sin(2\pi), sin(3\pi)[/tex], etc, are all in the image. But they are all 0! Does that mean that the image isn't a set, since 0 is in there many times? Not a bit. But the extra entries are redundant in this particular case.

    Another way to make this clearer is to think of the notation [tex]X = \{x_1, x_2, ...\}[/tex] as a shorthand for
    [tex]x_1 \in X, x_2 \in X, ...[/tex]
     
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