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is my proof that the union of countable sets is countable correct ?

If $A_1, A_2, A_3,\dots, A_n$ is a collection of countable sets, then the union $$A_1\cup A_2\cup A_3 \cup \dots A_n$$ is countable as well.

Proof. Base case: Consider the set $$B=A_2\setminus A_1$$ Clearly, $B\subseteq A_2$($B$ is countable) and $A_1\cup B$ = $A_1\cup A_2$.

If $B$ is finite, then $$B= \{b_1, b_2, b_3, b_4, \dots, b_j \}\quad j\in\mathbb{N}_0$$ and so we can construct a bijection $$f(n)=\begin{cases} b_n\quad n\leq j\\ a_{n-j}\quad n> j \end{cases}$$ If $B$ is infinite, then we can construct a bijection $$f(n)=\begin{cases} b_{\frac n2}\quad n\text{ even}\\ a_{\frac{n+1}{2}}\quad n\text{ odd} \end{cases}$$ Now, suppose the statement holds for $n= k\geq 2$, that is, $$A_1\cup A_2\cup A_3 \cup \dots A_k$$ is a countable set. Observe that $$(A_1\cup A_2\cup A_3 \cup \dots A_k)\cup A_{k+1}$$ is a union of two countable sets which, by the base case, is also countable. Thus, by induction, the statement holds for all $n\in\mathbb{N}.\qquad\square$

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    $\begingroup$ Your proof is okay if you are aiming to prove that a finite union of countable sets is countable (not an arbitrary union, as your title suggests). $\endgroup$ – drhab Mar 24 '18 at 16:35
  • $\begingroup$ Yeah the exercise was concerning finite unions. Thank you very much. :) $\endgroup$ – Adam Mar 24 '18 at 16:38
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Your proof is indeed valid for a finite union of countable sets. A shorter, more immediate proof goes as follows.

For any integer $n \gt 2$, one can construct a bijection $\phi$ between $\mathbb N$ and a finite union of $n$ countable sets $A = \bigcup_{i=1}^n A_i$

$$\phi : i \mapsto A_{i \mod n} \left (\left \lfloor \frac{i}{n} \right \rfloor \right )$$

Where $i \mod n$ is the rest of the euclidean division of $i$ by $n$, and $A_m(n)$ is the $n$-th element of the set $A_m$.

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  • $\begingroup$ It occurs to me that this is only valid for the union of disjoint sets. Will delete shortly, I can't on mobile $\endgroup$ – Matrefeytontias Mar 25 '18 at 10:39

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