\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}