Let $Σ = \{a, b, c\}$. Let Leven be the set of all even length strings in $Σ^*$
$(a)$ Construct a deterministic finite state automaton for $L_{even}$.
$(b$) We consider an operation $Erase_{ab}$ that takes as input a string $w \in Σ^*$ and erases all occurrences of the pattern $ab$ from $w$. Formally, it can be defined as follows:
Erase$_{ab}$(w):=$\left\{\begin{matrix} w &\text{if $w$ does not contain the pattern $ab$} \\ Erase_{ab}(w_{1})\:Erase_{ab}(w_{2}) & \text{if } w=w_1\:ab\:w_2 \text{ for some} w_1,w_2\in \Sigma^* \end{matrix}\right.$
For instance, $Erase_{ab}(cacb) = cacb$, $Erase_{ab}(cabcbab) = ccb$ and $Erase_{ab}(ab) = \epsilon$.
For a language $L$, we define $Erase_{ab}(L)$ to be the set of strings obtained by applying the $Erase_{ab}$ operation to each string in $L:$
$Erase_{ab}(L)$:= $\{ Erase_{ab}(w)\ |\ w\in L\}$
Show that $Erase_{ab}(L_{even}$) is a regular language.