We define a new regular operation called exclusive union $(\backslash$cup $)$

as follows, if A

and B

are languages then A$\backslash$cup $$B$=($A$\backslash$cup B$)($A$\backslash$cap B$)=\{$x$|$x in A

or x in B

$,$ but not both$\}$

$.$

a$)$ Using a DFA construction, prove that regular languages are closed under exclusive union. That is$,$ given two regular languages L$1$

and L$2$

$,$ prove that L$1\backslash$cup $$L$2$

is also a regular language $($do not do the induction, just the construction.$)$

b$)$ Illustrate your construction from a$)$ by applying it to the DFAs M$1$

and M$2$

$,$ which recognize L$1=\{$w in $\{0\}|$w$=0$k

where k$>=0$

is a multiple of $2\}$

and L$2=\{$w in $\{0\}|$w$=0$k

where k$>=0$

is a multiple of $3\}$

$,$ respectively.