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Math Help - Union of Two countably infinite sets

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    Union of Two countably infinite sets

    If A and B are both countably infinite then prove that A\bigcup B is countably inifinite.


    I was working on this problem with a friend and we know that this means that there is a bijection of both sets, but where do we go from there?
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    Quote Originally Posted by andirc5192 View Post
    If A and B are both countably infinite then prove that A\bigcup B is countably inifinite.


    I was working on this problem with a friend and we know that this means that there is a bijection of both sets, but where do we go from there?
    Since A and B are both countable we know that there exists a bijection from the integers to the sets.

    Let A = \{a_1,a_2,a_3, \cdots and
    Let B = \{b_1,b_2,b_3, \cdots

    let f:A \to \mathbb{N} \quad f(a_i)=i and
    let g:B \to \mathbb{N} \quad f(b_i)=i

    Now let h:A \cup B \to \mathbb{N}

    h(x)=\begin{cases}2i-1, \text{if } x \in A \\ 2i, \text{if } x \in B  \end{cases}

    Now show that this is both 1-1 and onto
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