Let L be any language. Let us define the transpose of L to be the language of exactly those words that are the words in L spelled backward. If w El, then reverse(w) El. For example, if

L = {a abb bbaab bbbaa}

then

transpose(L) = {a bba baabb aabbb}

(i) Prove that if there is an FA that accepts L, then there is a TG that accepts the transpose of L.
(ii) Prove that if there is a TG that accepts L, then there is a TG that accepts the transpose of L.
(iii) Prove that transpose(L₁L₂) = transpose(L₂)· transpose(L₁).