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Set notation

  1. May 18, 2009 #1
    I'm taking a class in abstract algebra this summer, so I thought I'd get ahead by reading the book before class starts.

    This is from a book called "Abstract Algebra: A Geometric Approach", chapter 1:

    Applying the Principle of Mathematical Induction with a slight modification.
    If [tex]S' \subset \{n \in N:n\geq n_0\}[/tex] has these properties:
    (1) [tex]n_0 \in S'[/tex]
    (2) If [tex]k \in S'[/tex] then [tex]k+1 \in S'[/tex]
    then [tex]S'=\{n \in N:n\geq n_0\}[/tex]
    If we define [tex]S=\{m \in N:m+(n_0-1) \in S'\}[/tex], we see that [tex]1 \in S[/tex] and [tex]k \in S[/tex], which leads to [tex]k+1 \in S[/tex] , and so [tex]S=N[/tex].
    Thus, [tex]S'=\{n \in N: n=n_0+(m-1)[/tex] for some [tex]m \in N\}=\{n \in N:n \geq n_0\}[/tex]

    I'm not sure how to interpret all that. I know the sideways U means "subset", and the sideways U with a line means "is an element of". But does something like this [tex]\{n \in N:n\geq n_0\}[/tex] mean n is an element of N only when [tex]n \geq n_0[/tex]?

    What about this: [tex]S=\{m \in N:m+(n_0-1) \in S'\}[/tex]?

    How do you interpret that?
     
  2. jcsd
  3. May 18, 2009 #2

    tiny-tim

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    Hi Bill! :smile:

    The : means "such that" …

    so that means "S is the set of all elements m of N such that m + n0 - 1 is an element of S´" :wink:
     
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