\begin{frame}
  \frametitle{Armstrong Axioms}

  \begin{goal}{}
    All implied FDs can be derived using the Armstrong axioms.
  \end{goal}
  \begin{block}{Armstrong axioms}
    \begin{itemize}
    \item \emph{Reflexivity:} 
      if $\beta \subseteq \alpha$, then $\alpha \to \beta$.
      
    \item \emph{Augmentation:}
      if $\alpha \to \beta$, then $\alpha \cup \gamma \to \beta \cup \gamma$.
  
    \item \emph{Transitivity:}
      if $\alpha \to \beta$ and $\beta \to \gamma$, then $\alpha \to \gamma$.
    \end{itemize}
  \end{block}
  \pause\medskip
  
  \begin{quiz}{\textwidth}{}
    Use the Amstrong axioms to show that
    \begin{enumerate}
      \item $\sql{isbn}           \to \sql{title}, \sql{publisher}$
      \item $\sql{isbn}, \sql{no} \to \sql{author}$
      \item $\sql{publisher}      \to \sql{publisherURL}$ 
    \end{enumerate}
    implies $\sql{isbn} \to \sql{publisherURL}$.
  \end{quiz}
  \pause
  
  \begin{exampleblock}{}
    \begin{enumerate}
      \item [4.] $\sql{title}, \sql{publisher} \to\sql{publisher}$ \hfill\iremark{by reflexivity}
    \pause
      \item [5.] $\sql{isbn} \to\sql{publisher}$ \hfill\iremark{by transitivity using 1. and 4.}
    \pause
      \item [6.] $\sql{isbn} \to\sql{publisherURL}$ \hfill\iremark{by transitivity using 5. and 3.}
    \end{enumerate}
  \end{exampleblock}
\end{frame}