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Math Help - Infinite sets and Countable sets

  1. #16
    jfk
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    Re: Infinite sets and Countable sets

    I'm sorry I still don't get it.

    I don't understand what (c) and (d) are and how do their elements look like.
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by jfk View Post
    I'm sorry I still don't get it.
    I don't understand what (c) and (d) are and how do their elements look like.
    For c). The characteristic function is {1_A}(x) = \left\{ {\begin{array}{rl}  {1,}&{x \in A} \\   {0,}&{x \notin A} \end{array}} \right..
    Look at the collection \left\{1_{\{k\}}:k\in\mathbb{N}\right\}

    For d) define f_k=\{(0,k),(1,k+1)\}. Now for each n\in\mathbb{N} the function f_n mapps \{0,1\}\to\mathbb{N}.
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    Re: Infinite sets and countable sets

    Quote Originally Posted by jfk View Post
    I don't understand what (c) and (d) are and how do their elements look like.
    If you need to visualize them to help you see the problem better, try visualizing them in terms of the suggestion made by DrSteve earlier in this thread.

    (c) Functions from \mathbb N to \{0,1\} are essentially infinite sequences of 0\text{'s} and 1\text{'s} e.g. (1,0,1,0,1,\ldots), (0,0,1,0,0,1,\ldots), (1,1,1,1,\ldots), etc. Can you visualize an infinite list of such infinite sequences?

    (d) Functions from \{0,1\} to \mathbb N can be regarded as sets of two ordered pairs of the form \{(0,m),(1,n)\} where m,n are natural numbers. Can you visualize an infinite list of such sets?
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    Re: Infinite sets and Countable sets

    To answer your question about irrational, algebraic and transcendental numbers:

    An algebraic number is the solution of a polynomial equation with integer coefficients. For example, the fraction m/n is a solution of nx-m=0. This shows that all rational numbers are algebraic numbers. But there are also lots of algebraic numbers that are irrational. For example, the polynomial equation x^2-c=0 gives two algebraic numbers that are irrational whenever c is not a perfect square. It is not hard to show that the set of algebraic numbers is countable. Therefore, there are uncountably many transcendental numbers.
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  5. #20
    jfk
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    Lightbulb Re: Infinite sets and Countable sets

    Thanks DrSteve, and thank you all for your time and your patience,
    I believe that the source of confusion came along with something I read somewhere, where it said that "all the Transcendental numbers are not Algebraic" therefore I concluded that all the Irrationals should be Transcendental however, now I understand my mistake since {x^2}-2=0 gives two irrational numbers that are not transcendental since they solve the last polynomial equation.

    by the way I figured out (c) and (d):

    (c) \{f\in\{0,1\}^\mathbb{N}|f(n)=\begin{cases}{1, n\in\{n\}} \\ {0, n\notin\{n\}}\end{cases}\}.
    This should generate a set with the following sequences: \{(1,0,0,0,...),(0,1,0,0,...),(0,0,1,0,...),(0,0,0  ,1,...),\}

    (d) \{f\in\mathbb{N}^{\{0,1\}}|f(\{n\})=\{(0,n),(1,n+1  )\}\}.
    This should generate a set which contains sets of the form: \{\{(0,1),(1,2)\},\{(0,2),(1,3)\},\{(0,3),(1,4)\},  ...\}

    Is that correct?
    Last edited by jfk; April 12th 2012 at 04:03 AM. Reason: added more content
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by jfk View Post
    Is that correct?
    The idea is correct, but the notation is a bit off.

    Quote Originally Posted by jfk View Post
    (c) \{f\in\{0,1\}^\mathbb{N}|f(n)=\begin{cases}{1, n\in\{n\}} \\ {0, n\notin\{n\}}\end{cases}\}.
    If f(n)=\begin{cases}{1, n\in\{n\}} \\ {0, n\notin\{n\}}\end{cases}\}, then f(n) = 1 for all n because n\in\{n\} for all n

    Quote Originally Posted by jfk View Post
    (d) \{f\in\mathbb{N}^{\{0,1\}}|f(\{n\})=\{(0,n),(1,n+1  )\}\}.
    There is a type error here because if f\in\mathbb{N}^{\{0,1\}}, then f cannot be applied to {n}.
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  7. #22
    jfk
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    Re: Infinite sets and Countable sets

    How can I generate the above sequences? I really spent a lot of time trying to understand this and I'm still stuck...
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    Re: Infinite sets and Countable sets

    Specifying a countable subset of some set X is more easily done by giving a sequence of elements of X (i.e., a function from \mathbb{N} to X) than by using a set-builder notation {x ∈ X | ...}. So a sequence of function f_0, f_1,\dots in \{0,1\}^{\mathbb{N}} can be defined by

    f_k(n)=\begin{cases}1& n=k \\0& n\ne k\end{cases}

    A sequence of function f_0, f_1,\dots in \mathbb{N}^{\{0,1\}} can be defined by

    f_k=\{(0,k),(1,k+1)\}
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    jfk
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    Re: Infinite sets and Countable sets

    f_k(n)=\begin{cases}1 & n=k \\0 & n\ne k\end{cases}
    But n and k are always equals, aren't they? That would lead every component in the sequence to be 1 as fa as I understand...
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by jfk View Post
    But n and k are always equals, aren't they?
    The notation

    f_k(n)=\begin{cases}1 & n=k \\0 & n\ne k\end{cases}

    means (in this context) that f is a function from \mathbb{N}\times\mathbb{N} to {0, 1}: f(k, n) = 1 or 0 depending on whether n = k. We write f(k, n) as f_k(n). The arguments (k, n) of f can be any pair from \mathbb{N}\times\mathbb{N}.
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  11. #26
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    Re: Infinite sets and Countable sets

    Thanks emakarov now I got it
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