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Cardnality of Infinite Sets

  1. Apr 29, 2008 #1
    1) Find the cardnality of the set of continuous functions from R to R.

    Let's consider a simplified version: for the cardnality of the set of functions from R to R, I can compare it with the set of characteristic functions of subsets of R and conclude that they both have cardnality 2c

    But when the word "continuous" is inserted, how can I find its cardnality?

    Thanks for any help!
     
  2. jcsd
  3. Apr 29, 2008 #2

    Dick

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    If f is a continuous function and you know the value of f(q) for all rational q, do you know the value of f for any real value?
     
  4. Apr 29, 2008 #3
    I think it would be hard to tell specifically. Since f is continuous, the values can't go too far, but there is still a wide possible range. Another thing is that we may not know the value of f(q) for q rational
     
  5. Apr 29, 2008 #4
    Also,
    since |{functions from R to R}|=2c,
    I think that |{continuous functions from R to R}|<2c.
    This narrows down the possible answers, but I still don't know how to get the cardnality exactly.
     
  6. Apr 29, 2008 #5

    Dick

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    That was a rhetorical question. Let me put it this way, if you know the value of f for all rationals, then you know the value of f for all reals. Why do I believe this? That means while a general functions have cardinality c^c the cardinality of continuous functions may be less. Is it?
     
  7. Apr 29, 2008 #6
    Sorry, I am lost for two reasons...

    The question asks for the cardnality of the set of continuous functions from R to R, so shouldn't we not be assuming any further detail. It didn't say that the value of f for all rationals are known. But perhaps I am misunderstanding the question...

    Also, why is the answer c^c? I haven't encountered the cardinal number c^c so I don't know what it means. Shoudn't the answer be less than or equal to 2^c?
     
  8. Apr 29, 2008 #7

    Dick

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    c^c=2^c. Wasn't that what you showed in the first part? What I'm saying is that continuous functions are DETERMINED by their values on the rationals Q. Q is countable.
     
  9. Apr 29, 2008 #8

    HallsofIvy

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