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\title{\bf Assignment 1 \\[2ex]
\rm\normalsize CS 581 \\ Due October 4, 2018}
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\paragraph{Problem 1 [10 points]}
For languages $A$ and $B$, let the {\em{\bf perfect shuffle}} of $A$ and $B$ be the
language:
\[ \{ w \;|\; w=a_1b_1a_2b_2...a_kb_k, \text{ where } a_1...a_k \in A \text{ and } b_1...b_k \in B, \text{ each } a_i,b_i \in \Sigma \}  \]
Show that the class of Regular languages is closed under {\em{\bf perfect shuffle}}.


\paragraph{Problem 2 [10 points]}
For languages $A$ and $B$, let the {\em{\bf shuffle}} of $A$ and $B$ be the
language:
\[ \{ w \;|\; w=a_1b_1a_2b_2...a_kb_k, \text{ where } a_1...a_k \in A \text{ and } b_1...b_k \in B, \text{ each } a_i,b_i \in \Sigma^* \}  \]
Show that the class of Regular languages is closed under {\em{\bf shuffle}}.

\paragraph{Problem 3 [10 points]}
For any string $w$, define
\[SCRAMBLE(w) = \{ t \;|\; t \text{ contains the same symbols as $w$ in any order}  \}\]
For any language $L$, define
\[SCRAMBLE(L) = \{ t \;|\; t \in SCRAMBLE(w) \text{ for some } w \in L \}\]
Prove or disprove that the if $L$ is a Regular language and $\Sigma = \{0, 1\}$ then $SCRAMBLE(L)$ is Regular.

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