Bijections from $A$ to $B$, where $A$ is the set of subsets of $[n]$ that have even size and $B$ is the set of subsets of $[n]$ that have odd size Let $A$ be the set of subsets of $[n]$ that have even size, and let $B$ be the set of subsets of $[n]$ that have odd size.  Establish a bijection from $A$ to $B$.  The following bijection is suggested for $n=3$:
$$\matrix{A: & \{1,2\} & \{1,3\} & \{2,3\} & \varnothing\\

B: & \{1,2,3\} & \{1\} & \{2\} & \{3\}}$$
I know that first we have to establish a function that is both surjective and injective so that it is bijective. I don't know where to take a step from here in the right direction. So I need a bit of guidance. 
Something suggested is let f be the piecewise function:
$$f(x) = x \setminus \{n\}\text{ if }n \in x\text{ and }f(x) = x\cup\{n\}\text{ if }n \notin x$$

For $n$ an odd positive integer, there is a natural procedure.  The mapping that takes any subset $E$ of $[n]$ with an even number of elements to its complement $[n]\setminus E$ is a bijection from $A$ to $B$.
Dealing with even positive $n$ is messier.  Here is one way. The subsets of $[n]$ can be divided into two types: (i) subsets that do not contain $n$ and (ii) subsets that contain $n$.  
(i) If $E$ is a subset of $[n]$  with an even number of elements, and does not contain $n$, map $E$ to $[n-1]\setminus E$.  Call this map $\phi$.
(ii) If $E$ is a subset of $n$ with an even number of elements, and contains $n$, map $E$ to $\{n\} \cup \phi^{-1}(E\setminus\{n\})$.     
