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Thread: Countability Proof

  1. #1
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    Countability Proof

    Fix $\displaystyle n >= 1$. Show that if A1,A2, . . . ,An are countable, then A1A2 . . . An is countable.

    Does this require knowledge of the fact that N x N is countably infinite?

    Thanks in advance.
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  2. #2
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    Hello,
    Quote Originally Posted by h2osprey View Post
    Fix $\displaystyle n >= 1$. Show that if A1,A2, . . . ,An are countable, then A1A2 . . . An is countable.

    Does this require knowledge of the fact that N x N is countably infinite?

    Thanks in advance.
    You can use the fact that $\displaystyle \mathbb{N}^n$ is countable. Which can be done by induction, and you'll have to use the fact that $\displaystyle \mathbb{N} \times \mathbb{N}$ is countable.

    1.
    You know that there exists an injective mapping $\displaystyle \phi_2 ~:~ \mathbb{N}^2 \to \mathbb{N}$
    Let $\displaystyle n \geqslant 2$
    Basis : for n=2, it's verified (I assume it's a know fact for you)
    Inductive hypothesis : assume that there exists an injective mapping $\displaystyle \phi_n ~:~ \mathbb{N}^n \to \mathbb{N}$
    Now, you have to prove that there exists an injective mapping $\displaystyle \phi_{n+1} ~:~ \mathbb{N}^{n+1} \to \mathbb{N}$
    For this, define $\displaystyle \phi_{n+1}$ this way :
    $\displaystyle \phi_{n+1}(x_1,\dots,x_n,x_{n+1})=(\phi_n(x_1,\dot s,x_n),x_{n+1})$
    It is easy to show that it's injective, knowing that $\displaystyle \phi_n$ is injective.


    2.
    Now prove that there exists an injection $\displaystyle \psi$ from $\displaystyle A_1 \times \dots \times A_n$ to $\displaystyle \mathbb{N}^n$
    Since $\displaystyle A_1,\dots,A_n$ are countable, there exist injections $\displaystyle \psi_i ~:~ A_i \to \mathbb{N}$
    Define $\displaystyle \psi (x_1,\dots,x_n)=(\psi_1(x_1),\dots,\psi_n(x_n))$
    Once again, it's easy to show that it's injective.


    3.
    $\displaystyle A_1 \times \dots \times A_n \stackrel{\psi}{\longrightarrow} \mathbb{N}^n \stackrel{\phi_n}{\longrightarrow} \mathbb{N}$
    We know that $\displaystyle \psi, \phi_n$ are injective.
    Now you just have to prove that the composite of two injective functions is injective, in particular $\displaystyle \phi_n \circ \psi$.

    And you're done.
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  3. #3
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    Quote Originally Posted by h2osprey View Post
    Fix $\displaystyle n >= 1$. Show that if A1,A2, . . . ,An are countable, then A1A2 . . . An is countable.

    Does this require knowledge of the fact that N x N is countably infinite?

    Thanks in advance.
    If you know that $\displaystyle \mathbb{N}\times \mathbb{N}$ is countable then it is easy to prove this result.

    If $\displaystyle |A| = |B|= |\mathbb{N}|$ then $\displaystyle |A\times B| = |\mathbb{N}\times \mathbb{N}|$ is countable.

    Thus, $\displaystyle |A_1\times ... \times A_n| = |\mathbb{N} \times ... \times \mathbb{N}|$.
    We established that $\displaystyle \mathbb{N}^2$ was countable.
    If $\displaystyle \mathbb{N}^k$, $\displaystyle k\geq 2$ is countable then $\displaystyle \mathbb{N}^{k+1} = \mathbb{N}^k \times \mathbb{N}$ is countable.
    The rest follows by induction.
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