Equivalence of uncountable sets

Let $A$ be an uncountable set and $B$ a countably infinite set. Find an injection $f: A \cup B \to A$.

Any ideas about the construction of such an injection?

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1 Answer

Let $C=\{c_1,c_2,c_3,\ldots\}$ be a countably infinite subset of $A$ (you need the axiom of choice to show that such a subset exists). Write $B\backslash A$ as $\{b_1,b_2,b_3,\ldots\}$ if it is infinite. Define $f:A\cup B\to A$ by letting $f(x)=x$ for $x\in A\backslash (C\cup B)$. Let $f(c_n)=c_{2n}$ and $f(b_n)=c_{2n-1}$. If $B\backslash A$ is finite $\{b_1,b_2,\ldots,b_m\}$, let $f(x)=x$ again for $x\in A\backslash (C\cup B)$, let $f(b_n)=c_n$ and let $f(c_n)=f(c_{n+m})$.

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Thank you very much! – Stefan Oct 7 '12 at 21:31
Minor nit: in the last sentence you should have $f(c_n)=c_{2n}$ rather than $f(c_n)=c_{2_n}$. – Rick Decker Oct 7 '12 at 23:39
There is a small potential issue here though, what if $A \cap B \neq \emptyset$? In that case $f$ is not well defined...The proof needs a minor change to address this ;) – N. S. Oct 8 '12 at 0:02