If $M$ is a DFA that recognizes language $B,$ you can manipulate that accept and non$-$accept states of $M$ to yield a new DFA, $M\text{'},$ that recognizes the complement of $B.$

A$.(10)$ Determine the manipulation you need to obtain $M^{\text{'}}$ :

B$.(10)$ Show that $M^{\text{'}}$ recognizes the complement of $B.$ You do not need an inductive or particularly formal proof, but you must convincingly argue it$.$

C$.(5)$ Conclude, formally, that the class of regular languages is closed under complement.