Consider the language L₆ = { w {a,b}^*: #_a(w) #_b(w)}.

1. Give a CFG for L₆.
2. Prove that, for any string w in L₆, w is generable from the start variable in your grammar. You can model your proof on the proof, given in class, that S aSb | bSa | SS | can generate every string over {a,b} with an equal number of a's and b's.
3. Prove using "induction for recursively defined objects" (concise form) that every string generated from the start variable in your grammar has at least as many a's as b's.
4. Is your grammar ambiguous? If so, give two distinct derivation trees for the same string. If not, argue briefly why your grammar is unambiguous.