"Prove the following: If A$,$ B are regular languages, then

$$ AR is also regular, where AR $=\{$wkwk$1...$ w$0|$w$0...$ wk$1$wk in A$\}.$

$$ A$\backslash$geoquad B is also regular, where A$\backslash$geoquad B $=\{$w$|$w $=$ a$1$b$1...$ akbk, where a$1...$ ak in A and b$1...$ bk in B$,$ each ai$,$ bi in

$\backslash$Sigma $\}.$

$$ extra credit: A$$B is also regular, where A$$B $=\{$w$|$w $=$ a$1$b$1...$ akbk, where a$1...$ ak in A and b$1...$ bk in

B$,$ each ai$,$ bi in $\backslash$Sigma $\}"$