\begin{frame}{Formal Languages}
  \begin{goal}{}
    A \emph{formal language} is a \emph{set of words}.
  \end{goal}
  \pause\bigskip
  
  \begin{block}{}
    A \emph{(formal) language} $L$ is a subset of $\Sigma^*$, that is, $L \subseteq \Sigma^*$.
  \end{block}
  Here $\Sigma^*$ is the set of all words over $\Sigma$.
  \pause\bigskip

  \begin{exampleblock}{}
    The set of all parseable C programs form a language.
  \end{exampleblock}
  \pause

  \begin{exampleblock}{}
    $\{\,ab,\; aab,\; bbaaabb\,\}$ is a finite language over $\Sigma = \{a,b\}$
  \end{exampleblock}  
  \pause
  
  \begin{exampleblock}{}
    $\{\,ab^n a \mid n \geq 1\,\}$ is an infinite language over $\Sigma = \{a,b\}$:
    \begin{talign}
      \{\,aba,\; abba,\; abbba,\; abbbba,\; \ldots\,\}
    \end{talign}
  \end{exampleblock}
  \pause
  
  \begin{exampleblock}{}
    $\{\, a^nb^n \mid n \geq 0 \,\}$ is an infinite language over $\Sigma = \{a,b\}$:
    \begin{talign}
    \{\,\lambda,\; ab,\; aabb,\; aaabbb,\; aaaabbbb,\; \ldots \,\}
    \end{talign}
  \end{exampleblock}
\end{frame}