\begin{frame} \frametitle{Database States} Let $S = (\, \{\,R_1, \dots, R_m\,\},\, \var{schema},\, C \,)$ be a database schema. \begin{block}{} A \textbf{database state} $I$ for database schema $S$ defines \begin{itemize} \item for every relation name $R_i$, \\ a finite \textbf{set of tuples} $I(R_i)$ with respect to $\schema{R_i}$ \end{itemize} \end{block} If $\schema{R_i} = (A_{1}:D_{1},\dots,A_{n}:D_{n})$, then \begin{talign} I(R_i) \subseteq \dom{D_{1}} \times \cdots \times \dom{D_{n}} \end{talign} Thus $I(R_i)$ is \emph{a relation in the mathematical sense}. \pause \begin{exampleblock}{} \emph{Databases state = set of tables conforming to the schema:} \begin{center}\vspace{-2.5ex} \scalebox{.5}{{\ttfamily\tableStudents\tableExercises}} \vspace{-1.5ex} \end{center} \emph{Except:} \begin{itemize} \item there is \emph{no order} on the tuples (rows), and \item tables contain \emph{no duplicate} tuples. \end{itemize} \end{exampleblock} \end{frame}