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