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Math Help - Decimal Expansion, 1-1 onto

  1. #1
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    Decimal Expansion, 1-1 onto

    Use the fact that every real number has a decimal expansion to produce a 1-1 function that maps S into (0,1). Discuss whether the formulated function is onto.


    S={(0,1):0<x, y<1}

    I don't even know where to begin. The whole decimal expansion business has me confused.
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    Quote Originally Posted by kathrynmath View Post
    Use the fact that every real number has a decimal expansion to produce a 1-1 function that maps S into (0,1). Discuss whether the formulated function is onto.


    S={(0,1):0<x, y<1}

    I don't even know where to begin. The whole decimal expansion business has me confused.
    I don't understand the notation you used to define S (x and y appear on the right side of the colon but not the left side.. ???), thus I don't know exactly what you're asking, but it looks related to this

    Cantor's diagonal argument - Wikipedia, the free encyclopedia
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    oh, the (0,1) in S should have been (x,y)
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    I'm sorry I can read the link, but don't really understand it. We haven't studied Cantor's Theorem yet.
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  5. #5
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    Quote Originally Posted by kathrynmath View Post
    oh, the (0,1) in S should have been (x,y)
    Ah, that makes more sense.

    I think we can just express x as its decimal expansion \,0.x_1x_2x_3\dots where \,x_1 is the first digit after the decimal point, similarly with y, then define \,z = x_1y_1x_2y_2\dots, that is f(x,y) = z. Note that with this function it is not possible to get, for instance, 0.09090909...
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    Ahh, that makes more sense.
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