Show that, if $\Omega$ is a finite set, then $\mathfrak{F}$ contains an even number of subsets of $\Omega$.
Proof: Let $f:\mathfrak{F}\rightarrow\mathfrak{F}$ such that $f(A)=\Omega\setminus A,\text{ }\forall A\in\mathfrak{F}$. Note that $f$ is bijective. That is, $\forall A\in\mathfrak{F},\text{ }\exists!\Omega\setminus A\neq A\in\mathfrak{F}$, so $\mathfrak{F}$ has an even number of subsets of $\Omega$.
Is this correct?