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Help with Set-Builder notation

  1. Jul 24, 2011 #1
    Hello. Say that I want to express a set consisting of the values of a continuous function, G(x), over a range of integers. Is this a valid/acceptable format?
    [tex] \mathbf{G} = \{G(x) : \forall (x \in \mathbb{Z})(x \in [-range, range])\} [/tex]

    or say that I want to express the vector formed by averaging m other vectors, all with n dimensions(indices). Would this express the operation correctly?

    [tex]
    \mathbf{\bar{S}} = \left\{ \dfrac1m \sum\limits_{i=1}^m \mathbf{S_i}_j : \forall \left(j \in \mathbb{Z}\right)\left(j \in [1,n]\right) \right\}
    [/tex]

    Thanks for any help!
     
  2. jcsd
  3. Jul 25, 2011 #2
    For the 1st one the set builder notation is:

    G = {g(x) : xε(-range, range)} or G = {g(x): g:[-range,range] => R]}

    Now since the rang of g is a set of integers ,then by definition g is continuous

    For the 2nd problem : Do you mean that you want to express the average sum of :

    [tex]a_{1}+a_{2}+.........a_{m}[/tex] vectors ,where ,

    [tex] a_{1}[/tex]= ([tex] x_{11},x_{12},...........x_{1n}[/tex])??
     
  4. Jul 25, 2011 #3
    So how do I make it clear that the set being defined in the first problem is only the values of G when it is given an integer argument? Ie, I want to take a continuous function and use it to generate a set by giving it only certain inputs. Think of the set I'm trying to build as some discrete pixelated approximation of G.

    I want to take the average of a group of vectors by averaging their components at a series of posititions. For example, this operation of the vectors (1,3,5) and (2,4,6) would yield (1.5, 3.5,5.5)
     
  5. Jul 25, 2011 #4
    Let us take things one by one >

    G = {[tex]g(x) : x\in Z\wedge -a\leq x\leq a, g(x)=x^2[/tex]} ,where a is an integer

    I just used an example of a continuous function
     
    Last edited: Jul 25, 2011
  6. Jul 25, 2011 #5

    Hurkyl

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    There are three basic ways to use set-builder notation to define a set:
    • [tex]\{ x \in A \mid P(x) \}[/tex]
    • [tex]\{ f(x) \mid x \in A \}[/tex]
    • [tex]\{ f(x) \mid x \in A \wedge P(x) \}[/tex]
    where A is a set, f is a function, and P is a predicate. (all three are allowed to involve variables other than x)

    Key syntactic features of these are:
    • x is introduced as a dummy variable. (Similar to how x appears in expressions like [itex]\int_a^b f(x) \, dx[/itex])
    • All forms include something that "binds" x to range over a particular set.
    • Forms 2 and 3 require an expression in x on the left side, and forms 1,3 involve a predicate in x on the right hand side

    A lot of variations are allowed, so long as it's clear what is meant. (e.g. I might often write a comma instead of [itex]\wedge[/itex] in the third form, and I might omit the [itex]x \in A[/itex] part if it can be inferred from the rest of the expression)


    The notation you were using is flawed because your right sides aren't expressions in the dummy variable. It's a similar grammatical error as something like:
    [tex]f(x) = \int_0^1 x^2 \, dx[/tex]​
    which should be avoided because the x appearing in the integral cannot be the x appearing in the left hand side.

    Of course, the above equation is technically a valid one that asserts [itex]f(x) = 1/3[/itex].



    I think your thinking was that you wanted to express that you were defining the set of all things described by the notation -- however that is already what the set-builder notation means. The third form, for example, reads as
    The set of all f of x where x comes from A and satisfies P​

    But what you wrote unfortunately introduces a new dummy variable. Your expression
    [tex]\mathbf{G} = \{G(x) : \forall (x \in \mathbb{Z})(x \in [-range, range])\}[/tex]​
    means exactly the same thing as
    [tex]\mathbf{G} = \{G(x) : \forall (y \in \mathbb{Z})(y \in [-range, range])\}[/tex]​
    Technically this is well-defined (keeping in mind the acceptable variations on notation), but it defines G to be the empty empty set, because your expression on the right hand side is identically false.
     
  7. Jul 26, 2011 #6
    Ah, I understand now. Had I not used the "for all" symbol, would my expressions be acceptable? Or do paired products of parenthetical arguments not imply logical conjunction?
     
  8. Jul 26, 2011 #7

    Hurkyl

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    It's fine, I think, but it really does look odd -- I expect most people would understand what you meant, even if they find it odd or think you made a typographical error.


    I imagine it would be much more acceptable in certain contexts -- e.g. in a book that has already adopted the convention to write "logical and" as an implied product.
     
  9. Jul 26, 2011 #8
    Fair enough-- the notation I was using was from Wikipedia, so I'd expect a little bit of irregularity. I'll switch to the standard conjunction from now on.

    I'm assuming that the vertical bar and colon are interchangeable, although please let me know if they are not for any reason. The bar reminds me a little bit too much of the "or" operator in programming, hence my avoidance of it.

    Thanks very much for your help. This was all great advice.
     
  10. Jul 26, 2011 #9

    Hurkyl

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    I view them as interchangeable, but I really have no idea how widespread that view is.
     
  11. Jul 26, 2011 #10
    They are interchangeable; they're just different notations for the same thing.
     
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