I've been stuck on this question and I'm confused as to how to approach it:
Let $G$ be a set defined as follows:
- if $x$ is a propositional variable, then $x \in G$;
- if $f_1,f_2 \in G$, then $\lnot f_1 \in G$, and $(f_1 \land f_2) \in G$;
- nothing else belongs to $G$.
For a formula $f \in G$, let $c_{not}(f)$ be the number of occurrences of $\lnot$ in $f$, and $c_{and}(f)$ be the number of occurrences of $\land$ in $f$. Let $H = \{f \in G : c_{not}(f) = c_{and}(f)\}$. That is, $H$ is the set of formulas in $G$ with equal number of $\lnot$'s and $\land$'s.
Prove that for any formula $f \in G$, there is a formula $f'$ such that $f' \in H$ and $f'$ and $f$ are logically equivalent.