Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

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 $f(c_n)=f(c_{n+m})$.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.