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Math Help - Infinite sets and Countable sets

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    jfk
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    Question Infinite sets and Countable sets

    Hi everybody,

    I need to prove the following sets are infinite sets by finding a countable in everyone of them:
    a) \mathbb{Q}\backslash\mathbb{Z}
    b) \mathbb{R}\backslash\mathbb{Q}
    c) \{0,1\}^\mathbb{N}
    d) \mathbb{N}^{\{0,1\}}

    How can I do this?
    I was wondering if the following would be correct:
    for example in (a): \mathbb{A}\subset\mathbb{Q}\backslash\mathbb{Z},
    where \mathbb{A}= \{ x | x = \frac{1}{n}, & \forall  n\in \mathbb{N} \}.
    Thanks in advance for the help.
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by jfk View Post
    I was wondering if the following would be correct:
    for example in (a): \mathbb{A}\subset\mathbb{Q}\backslash\mathbb{Z},
    where \mathbb{A}= \{ x | x = \frac{1}{n}, & \forall  n\in \mathbb{N} \}.
    This is almost correct except that 1 ∈ A and, depending on the definition of natural numbers, 0 may be a natural number. Also, it is more correct to write this version of A as \{x\mid\exists n\in\mathbb{N}\;x=1/n\} or \{1/n\mid n\in\mathbb{N}\}.

    Why don't you try other parts and post the results for verification?
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    Re: Infinite sets and Countable sets

    Thanks emakarov I just realized that 1 and 0 are problematic in my definition of A.
    I'll think about (b)(c)(d) and I'll post them later...
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    Re: Infinite sets and Countable sets

    For (b) I'm not sure about that one. If I take out all the rationals from the real numbers, then I'll have only the transcendentals in that remaining set and therefore it cannot be counted... Is that right?
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by jfk View Post
    to prove the following sets are infinite sets by finding a countabl(y infinite subset) in everyone of them:
    b) \mathbb{R}\backslash\mathbb{Q}.
    Have you thought about \left\{ {\frac{\pi }{{{2^n}}}:~n \in \mathbb{N}} \right\}~?
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    Re: Infinite sets and Countable sets

    ouch! Nope I forgot about that. Thanks
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    Re: Infinite sets and Countable sets

    @jfk Just a correction to something you said:

    Not every irrational number is transcendental. For example square roots of non-perfect squares are irrational, but not transcendental.

    Hint for (c): functions from the naturals to {0,1} are essentially just countable sequences of 0's and 1's. Can you write down infinitely many such sequences? There are many ways to do this.

    Hint for (d): elements of this set are essentially ordered pairs of natural numbers. Can you write down infinitely many such pairs?
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by DrSteve View Post
    Not every irrational number is transcendental. For example square roots of non-perfect squares are irrational, but not transcendental.
    10x DrSteve for the correction. Though I'm not sure I understood your example. I'm pretty confused about the relation (e.g: who contains who) between Irrational, Algebraic and Transcendental numbers, I would apreciate it very much if some one can help me make some order with those concepts.
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    Re: Infinite sets and Countable sets

    Why don't you read Wikipedia about it?
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    Re: Infinite sets and Countable sets

    I'm sorry I still don't understand (c) and (d), what's the meaning of a set to the power of another set?
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by jfk View Post
    I'm sorry I still don't understand (c) and (d), what's the meaning of a set to the power of another set?
    {\left\{ {0,1} \right\}^\mathbb{N}} is the set of all functions mapping \mathbb{N}\to\{0,1\}.
    Think characteristic functions

    {\mathbb{N}^{\left\{ {0,1} \right\}}} is the reverse of that.
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    Re: Infinite sets and Countable sets

    Ok then,

    (c) could be \left\{f|f(x) = \left\{ \begin{array}{rcl}1 & \mbox{for x is Even} \\ 0 & \mbox{for x is Odd}\end{array}\right.\right\} ?
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by jfk View Post
    Ok then,

    (c) could be \left\{f|f(x) = \left\{ \begin{array}{rcl}1 & \mbox{for x is Even} \\ 0 & \mbox{for x is Odd}\end{array}\right.\right\} ?
    That would be a single function in the set, not a countabe collection of them.
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    Re: Infinite sets and Countable sets

    What about \{f\mapsto\(f(1),f(2),f(3),...\)\}???
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    Re: Infinite sets and Countable sets

    Quote Originally Posted by jfk View Post
    What about \{f\mapsto\(f(1),f(2),f(3),...\)\}???
    that's not good, either. what about a function f for which f(k) = 1, and f(n) = 0, if n ≠ k? how many of THOSE functions are there?
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