I have a context-free grammar defined by the production S:
S → aSbS ∣ bSaS ∣ ε
I need to prove that the CFG "G" can be defined as a language L(G) where
L(G) = {w ∈ {a, b}∗ ∶ na(w) = nb(w)}.
Where na(w) = number of a's in w, and nb(w) is the number of b's in w
How can I go about proving something like this? Is there a method?
Without giving the answer away.
Let us denote $L'$ the language defined by the CFG $S$.
To prove $L'$ = $L$ you must do two things:
prove $w' \in L' \Rrightarrow w'\in L(G)$, meaning $L' \subseteq L(G)$
prove $w \in L(G)\Rrightarrow w\in L'$, meaning $L(G) \subseteq L'$
Combine proofs 1 and 2, and you get that $L' = L(G)$.
The proofs themselves are not hard, since $L'$ has a very specific structure.
