In mathematics, fuzzy sets are sets whose elements have \textit{degrees} of membership, described by a \textit{membership function}\cite{buckley2002introduction}.
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\begin{itemize}
\item Degree of membership is defined in interval\footnote{In theory, it could be higher than 1, but in practice it is almost never used}$[0, 1]$
\item Elements can have different degree of membership to different fuzzy sets
\item If the uncertainty is not handled, we talk about \textbf{type-1} fuzzy sets, \textbf{type-2} otherwise
\end{itemize}
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\begin{frame}{Fuzzy Set Interpretation}
How do we represent \textit{numerical} value in a fuzzy set? With the use of \textit{linguistic variables}\cite{lieb1993linguistic}, \textbf{not} probabilities.
\caption{Example interpretation of fuzzy sets. At the given temperature point, we can tell that the measured medium is "not hot", "slightly warm" and "almost cold". It does not mean that the chance the water is cold is 75\%.
Let $U$ be the \textit{universe of discourse} and $x$ be the element in it. The \textit{membership function}$f^A$ assigning \textit{degree of membership}$\mu_A$:
It is no coincidence, that these truth tables for binary fuzzy sets are identical to their Boolean counterparts\footnote{DeMorgan's law, associativity, comutativity and distributivity also apply.}.
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\begin{frame}{Triangular Norm (T-norm)}
A T-norm is a \textbf{continuous} function $T:[0,1]\times[0,1]\rightarrow[0,1]$, satisfying these axioms:
\begin{itemize}
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\item[Neutrality\footnote{Also referred to as a \emph{boundary condition}.}]$T(a, 1)= a$
The fuzzy control systems are commonly used \cite{ross2009fuzzy} where there are not enough resources for highly advanced systems like \textbf{PID\footnote{Proportional-integral-derivative} controller}, \textbf{Artificial neural network} or \textbf{Genetic algorithm}\cite{rajasekaran2003neural}.
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\subsection{Software}
\begin{frame}{MATLAB Fuzzy Toolbox Introduction}
\begin{itemize}
\item Provides a complete set of functions to design an implement various fuzzy logic processes \cite{sivanandam2006introduction}
\item Major fuzzy logic operation-fuzzification, defuzzification, and the fuzzy inference
\item Can be implemented using the Graphical User Interface (GUI)
\end{itemize}
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\begin{frame}{MATLAB Fuzzy Toolbox}
Features:
\begin{itemize}
\item It provides tools to create and edit fuzzy inference system (FIS).
\item Allows integrating fuzzy systems into simulation with Simulink.
\item It is possible to create stand-alone C programs that call on fuzzy systems
%\item Use \texttt{tabular} for Basic Tables! --- See Table~\ref{tab:widgets}, for Example.
%\item You Can Upload a Figure (JPEG, PNG or PDF) Using the Files Menu.
%\item to Include It in Your Document, Use the \texttt{includegraphics} Command (See the Comment Below in the Source Code).
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%\begin{table}
%\centering
%\begin{tabular}{l|r}
%Item & Quantity \\\hline
%Widgets & 42 \\
%Gadgets & 13
%\end{tabular}
%\caption{\label{tab:widgets}An example table.}
%\end{table}
% Let $X_1, X_2, \ldots, X_n$ be a sequence of independent and identically distributed random variables with $\text{E}[X_i] = \mu$ and $\text{Var}[X_i] = \sigma^2 < \infty$, and let
% $$S_n = \frac{X_1 + X_2 + \cdots + X_n}{n}
% = \frac{1}{n}\sum_{i}^{n} X_i$$
% denote their mean. Then as $n$ approaches infinity, the random variables $\sqrt{n}(S_n - \mu)$ converge in distribution to a normal $\mathcal{N}(0, \sigma^2)$.
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% \begin{block}{Examples}
% Some examples of commonly used commands and features are included, to help you get started.