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Math Help - Z and Q are countable

  1. #1
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    Z and Q are countable

    I'm reading the text about the proof of Z and Q are countable.

    The proof reads:

     \mathbb {Z} is the union of the countable sets  \mathbb {N} \ , \ \{ -n : n \in \mathbb {N} \} \ , \ \{ 0 \} , and one can define a surjection  f: \mathbb {Z} ^2 \rightarrow \mathbb {Q} by  f(m,n) = \frac {m}{n} if  n \neq 0 and  f(m,0) = 0

    End of proof.

    I understand that basically the system of Z and Q are equivalent, so if Z is countable so is Q, but where in the proof did it show that there exist an injection from Z to the set of natural numbers? Thanks.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    I understand that basically the system of Z and Q are equivalent, so if Z is countable so is Q, but where in the proof did it show that there exist an injection from Z to the set of natural numbers? Thanks.
    f:\mathbb{Z} \mapsto \mathbb{N}\;,\,f(n) = \left\{ {\begin{array}{rl}<br />
   {2n,} & {n \geqslant 0}  \\<br />
   {2\left| n \right| + 1,} & {n < 0}  \\<br /> <br />
 \end{array} } \right.
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  3. #3
    Super Member Matt Westwood's Avatar
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    There's a theorem to the effect that the union of a countable number of countable sets is itself countable:

    Union of Countable Sets - ProofWiki
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