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\section*{Homework 2}

\begin{enumerate}
\item Review the first handout about sets of strings and read
      the second handout. Assuming the alphabet is $\{a, b\}$,
      decide which of the following equations are true in
      general for arbitrary languages $A$, $B$ and $C$:

\begin{eqnarray}
(A \cup B) @ C & =^? & A @ C \cup B @ C\nonumber\\
A^* \cup B^*   & =^? & (A \cup B)^*\nonumber\\
A^* @ A^*      & =^? & A^*\nonumber\\
(A \cap B)@ C  & =^? & (A@C) \cap (B@C)\nonumber
\end{eqnarray}

\noindent In case an equation is true, give an explanation;
otherwise give a counter-example.

\item What is the meaning of a regular expression? Give an
      inductive definition.

\item Given the regular expressions $r_1 = \epsilon$ and $r_2
      = \varnothing$ and $r_3 = a$. How many strings can the
      regular expressions $r_1^*$, $r_2^*$ and $r_3^*$ each
      match?


\item Give regular expressions for (a) decimal numbers and for
      (b) binary numbers. (Hint: Observe that the empty string
      is not a number. Also observe that leading 0s are
      normally not written.)

\item Decide whether the following two regular expressions are
      equivalent $(\epsilon + a)^* \equiv^? a^*$ and $(a \cdot
      b)^* \cdot a \equiv^? a \cdot (b \cdot a)^*$.

\item Given the regular expression $r = (a \cdot b + b)^*$.
      Compute what the derivative of $r$ is with respect to
      $a$, $b$ and $c$. Is $r$ nullable?

\item Prove that for all regular expressions $r$ we have
      
\begin{center} 
  $\textit{nullable}(r) \quad \text{if and only if} 
  \quad [] \in L(r)$ 
\end{center}

  Write down clearly in each case what you need to prove and
  what are the assumptions. 
  
\end{enumerate}

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