This is a problem proposed in the class. I hope I managed to capture its spirit.

Let $L_1,L_2$sub${}^{**},$ where $=\{0,1\}$ be regular languages. Consider $w_{1}in{L}_1,w_{2}in{L}_2$ of equal lengths $($if

of unequal length, after padding one of the strings to the right by $0\text{'}$s$,$ make them of equal length$),$ define

$L=\{\begin{array}{c}w=w_1\end{array}$ op op op $\{$:$b_k)|w_1=a_{1}cdots{a}_{k}in{L}_1,w_2=b_{1}cdots{b}_{k}in{L}_2$ as follows. In each

case, prove or disprove that $L$ is regular.

$($a$)$ op is element$-$wise standard exclusive on $$;

$($b$)a_i$ op $b_i$ is $a_i{b}_i$ if $a_i{b}_i$ otherwise $,$ i$.$e$.,$ unequal elements at a position is kept and equal elements are

cancelled out.

$($c$)a_i$ op $b_i$ is $a_i{b}_i$ if $a_i{b}_i$ otherwise $a_i,$ i$.$e$.,$ only one copy is kept in case of equal elements at a position.

$($d$)a_i$ op $b_i$ is $a_i{b}_i$ if $a_i=b_i$ otherwise $.$