"Let L$1,$ L$2\backslash$Sigma $,$ where $\backslash$Sigma $=\{0,1\}$ 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 $0$s$,$ make them of equal length$),$ define L $=\{$w $=$ w$1$ op w$2=($a$1$ op b$1)($ak op bk$)|$ w$1=$ a$1$ ak in L$1,$ w$2=$ b$1$ bk in L$2\}$ as follows. In each case, prove or disprove that L is regular.

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

$($b$)$ ai op bi is ai bi if ai $=$ bi otherwise $\backslash$lambda $,$ i$.$e$.,$ unequal elements at a position is kept and equal elements are cancelled out.

$($c$)$ ai op bi is ai bi if ai $=$ bi otherwise ai$,$ i$.$e$.,$ only one copy is kept in case of equal elements at a position.

$($d$)$ ai op bi is ai bi if ai $=$ bi otherwise $\backslash$lambda $."$