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Textbook error relating to uncountability?

  1. Jan 20, 2012 #1
    I am reading a textbook about communications systems, and just came across a part that I believe to be wrong. As shown below, it says that if you take a vector g = [g(t1) g(t2) ... g(tN)] and take the limit as the number of samples goes to infinity, the result is a continuous function of t. Wouldn't this imply a bijection between the natural numbers and a continuous interval which was disproven by cantor's second theorem?

    bij1.gif

    bij2.gif
     
  2. jcsd
  3. Jan 20, 2012 #2

    Stephen Tashi

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    Informal statements are often saved from outright errors by being vague. That passage did not give a definition for [itex] lim_{N \rightarrow \infty} \mathbb{g} [/itex].

    The definition of a limit of a sequence of vectors (which can be regarded as a sequence of function) cannot be deduced from the definition of the type of limit used in calculus, which deals with the limit of a single function at a given point. There are actually many different types of limits that are defined for sequences of functions. I'd have to look these up and re-study them before I could swear that none of them make the book's statement true.

    You are correct that the infinite sequence of discrete time samples does not ever become an uncountable set.
     
  4. Jan 20, 2012 #3

    pwsnafu

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    A "sequence of functions" would require the domains to be the same though...
    The only option I could think is define the sequence of sequences
    [itex]h = [g(t_1), \ldots, g(t_N), 0, 0, \ldots][/itex]
    and then look at limits. Err.

    The book really should have used step functions.
     
  5. Jan 21, 2012 #4

    Stephen Tashi

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    Yes, by many definitions it would. But statisticians say things like "For large N the binomial distribution can be approximated by a normal distribution". That's a statement connecting a function with a discrete domain to a function with a continuous domain. I don't know if people bother to formulate that type of approximation as a theorem about a limit of a sequence of functions, but I think if they wanted to, they could come up with something. There are so many possibilities for making up definitions!
     
  6. Jan 21, 2012 #5

    AlephZero

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    This a nice example of the difference between math and engineering.

    In engneering, any "real world" signal will be bandwith limited, and the statement in the book is true for bandwidth-limited signals.

    But I agree that the book's "explanation" isn't much more than an arm-waving reason why that seems plausible.
     
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