Let $F$ be the function that sends a finite subset of naturals to the set $\{\text{even}, \text{odd}\}$, according to the parity of its size. Can $F$ be extended to a $G$ on all subsets, so that for any (potentially infinite) set $A$ and element $a$ of $A$, we have the following?
$$G(A-\{a\})=\begin{cases}
\text{even}, & \text{if}\; G(A)=\text{odd} \\
\text{odd}, & \text{if}\; G(A)=\text{even}
\end{cases}$$
Transfinite induction doesn't seem to work.
Best Answer
Here is a (non-canonical) construction:
If $A,B$ are infinite subsets of $\mathbb N$, we say that they are equivalent if they differ by a finite set. That is, if the set of natural numbers which are in one but not both is finite. If we think of such subsets as identifying real numbers in $(0,1]$ then equivalence means that the two decimals coincide from some point on.
Choose a representative of each equivalence class and assign its parity arbitrarily (for example, say that all the representatives are Even). Then extend the notion of parity to the whole class via the given rule.
Note that this notion of parity does not appear to extend to infinite subsets. That is to say, if $B\subset A$ and $A,B$ are both infinite and the parity of both are known, then we can't infer the parity of the complement, $A-B$. My sense is that there is no notion of parity which works in this sense, but I am not sure.