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Thread: Fractals & Chaos!

  1. #1
    Dec 2011

    Fractals & Chaos!

    Hi all,
    I am in my final year at University doing a horrible module called Fractals & Chaos (really it's called Applied Mathematics - even though I find it to be purely pure mathematics)

    Now my main problem is the proofs of the module.. turns out it's more proof heavy than I ever imagined and my exam is very soon .. Shortly after the weekend.

    First problem I have really is:
    What is the image of a function - I've got the definition but I don't understand it. I am terrible at Set Theory most the time as I have never been taught it properly.

    Next question is:
    Open and Closed Sets - how can I find out when sets are open or closed? I sort of understand how to find the boundary of sets but beyond that I get a bit confused. And most of the questions involve a ball of radius epsilon.

    I'm unsure if this topic should be in Set Theory - I've put it in here instead because the course isn't all Set Theory. Thanks.
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  2. #2
    MHF Contributor

    Mar 2011

    Re: Fractals & Chaos!

    the image set of a function is the set of all images of the function. that probably didn't help any, huh?

    ok, if we have a function $\displaystyle f:A \to B$, and a is an element of A, then f(a) is called the image of a under f. for example if A = B = R, the real numbers,

    then one possible function is $\displaystyle f:x \to x^2$. so the image of -4 is 16, for example. and f(R) is the set of all squares in R, that is, all non-negative real numbers

    (since 0 or any positive real number has a square root). the image set is often written f(A), and is formally:

    $\displaystyle im(f) = f(A) = \{b \in B: \exists \ a \in A: f(a) = b\}$

    in plain english, the set of all values that f "hits" or "takes on". a function "does" something to each element of its domain, the image set reflects the result of that (another name for the image set is the range of f, although this usage is not 100% consistent, as some people mean "the co-domain", that is, all of B).

    f(A) will always be a subset of B, if f(A) = B, the function is called "onto", or surjective.

    the description of "open sets" and "closed sets" is hard to do in intuitive terms. since you seem a bit more comfortable with the concept of "boundary", i will try to phrase "open" and "closed" in terms of boundary points.

    first, it is helpful to think of a set S, as lying inside some larger set U (called a universe). then everything not in S, is called the complement of S in U (also written U-S). so the boundary points of S, are those points that are close to S AND U-S (a neighborhood of a boundary point intersects both sets), they are "on the fence (boundary)".

    the points in S that are NOT boundary points, are interior points (totally inside S). for these points, we can draw a (perhaps very tiny) circle (or ball, in higher dimensions) around our point without "leaving S" (intersecting U-S).

    so S open = all interior points = no boundary points.
    S closed = interior points (maybe, S might be "all boundary") + all boundary points.

    it's easier to imagine what a "typical open set" looks like, by imagining such sets on the real line. the "prototypical open set" is an open interval (a,b), that is:

    a < x < b. note that if x is close to b than a, but less than b, the interval:

    (x-(b-x)/2, x+(b-x)/2) is totally within this interval (draw a picture).

    a similar sub-interval can be constructed if x is closer to a, or half-way between a and b.

    now, the boundary points are {a}, and {b}. if we include these, we get the CLOSED interval:

    a ≤ x ≤ b, or [a,b].

    this is the basic idea, but in "more dimensions". in 2 dimensions, we use disks (filled in circles), instead of intervals, in 3 or more dimensions, we use "balls" (because the boundaries of these sets, are points all at a set distance from the center of the ball). this lets us reduce questions of "inside/outside" to questions about "how far away" (where we can use the wealth of knowledge we have about real numbers).
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  3. #3
    Dec 2011

    Re: Fractals & Chaos!

    Thank you very much for the reply ..
    It helped to an extent ^_^
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