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:
$f(x) = x \setminus \{n\}\text{ if }n \in x\text{ and }f(x) = x\cup\{n\}\text{ if }n \notin x$