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Math Help - Real Analysis / Cantors Diagonalisation possibly? S is the set of f:N->{0,1,2}

  1. #1
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    Real Analysis / Cantors Diagonalisation possibly? S is the set of f:N->{0,1,2}

    Hey all, i've come across a problem i'm stumped on:

    Let S be the set of all functions u: N -> {0,1,2} Describe a set of countable functions from S

    We're given that v1(n) = 1, if n = 1 and 2, if n =/= 1

    The function above is piecewise, except i fail with latex


    To begin with, im not exactly sure what the question is asking, are we looking for all functions u that map the natural numbers to either 0,1,2 since i imagine there would be uncountably many of these? Or do i need to write each u(n) as a decimal expansion using the numbers 0,1,2? To put it plainly, i'm very confused about what the question is asking so a point in the right direction would be much appreciated!

    So while i realise i haven't had a proper attempt at a solution, with a nudge in the right direction hopefully i can get on my way and ask for some assistance if/when i need it showing all relevant work i've done.

    Thanks everybody!
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  2. #2
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    I think there's some information missing here. There are lots of sets of countable functions from S. What does v1 have to do with anything? It's just one specific function from S. Why do you mention it?
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  3. #3
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    Apologies, the question just states V1 is an example of an element in S and to perhaps similarly describe a countable set of functions from S.

    I dont think there's anything special about it, but i'm providing all the information which came with the question incase something is of importance.
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  4. #4
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    OK. Like I said there are infinitely many (in fact, uncountably many) possible answers. Here's one:

    Let v_k(n)=1 if n\leq k, and v_k(n)=2 if n>k.

    Then \{ v_k|k\in \mathbb{N}\} is a countable set of functions from S.
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