Re: Peano's space-filling curve

From: Daniel Grubb (grubb_at_lola.math.niu.edu)
Date: 07/14/04

  • Next message: John Morgan: "Re: Peano's space-filling curve" 
    Date: 14 Jul 2004 14:24:04 GMT

    >> A is a set; that means it has some things that we
    >> call elements, some > things are elements of A
    >> and some things are not. That's all you need to know.

    >I can conceive of this, but then I encounter a new problem.
    >It always seemed to me that the domain of a function and
    >the rule that linked it to the codomain were inextricably
    >linked, with the set comprising the domain determining which
    >rules were applicable, while a given rule might only be
    >relevant to a certain class(es) of sets. If the domain was a
    >set of the names of famous people, then the rule 'has a
    >birthday on' can map the name to a codomain of 366 Julian
    >days, while the rule 'multiply by two' is meaningless.
    >However, its inverse 'multiply by 1/2' is not incompatible
    >with the codomain. Similarly the first mentioned rule is
    >meaningless when applied to a domain of integers while its
    >inverse 'is the birthday of' can be applied to integers, as
    >long as these represent Julian days.

    OK. Maybe you are ready for the formal definition of a function.
    Let A and B be sets. Any sets at all. A function f:A->B is
    a subset f\subsetof AxB with the following two properties:

    1) If a \in A, then there is a b \in B with (a,b) \in f.
    2) If a \in A, and b_1 ,b_2 \in B with (a,b_1 ) \in f and (a,b_2) \in f
    then b_1 =b_2.

    In other words, a function f is a collection of ordered pairs
    so that every a \in A has exactly one b\in B with (a,b)\in f.
    We give this unique b the name f(a).

    In essence, this identifies the intuitive idea of a function with
    its graph.

    --Dan Grubb

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