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\begin{document}
Exercises from fuzzy propositional and set operations.

\begin{exercise}
\noindent\par
\begin{enumerate}
\item 
Verify that operation $\orf.$ defined by
$$\alpha\orf.\beta = \sqrt{\alpha^2+\beta^2-\alpha^2\,\beta^2}$$
is a fuzzy disjunction.

\item 
Find the fuzzy conjunction dual to $\orf.$ with respect to the fuzzy negation
\begin{equation}
\notf. \alpha = \sqrt{1-\alpha^2} \,. \label{eq.neg}
\end{equation}
\end{enumerate}
\end{exercise}

\begin{exercise}
Fuzzy subsets $A,B$ of $\R$ have membership functions
\begin{align*}
 \mu_A(x) &= 
 \begin{cases}
 x-1\,, & 1<x<2\,,\\
 1\,, & 2\le x\le3\,,\\
 \frac{5-x}2\,, & 3<x<5\,,\\
 0, & \text{otherwise.}
 \end{cases}
 \\
 \mu_B(x) &= 
 \begin{cases}
 \frac x3\,, & 0<x<3\,,\\
 1\,, & 3\le x\le4\,,\\
 \frac{6-x}2\,, & 4<x<6\,,\\
 0, & \text{otherwise.}
 \end{cases}
\end{align*}
Find 
\begin{enumerate}
\item 
$\dop A^.$, where $\notf.$ is the fuzzy negation from~\eqref{eq.neg},
\item 
$A \capf P B$,
\item 
$A \cupf L B$.
\end{enumerate}
Describe the results by formulas and draw their graphs.
\end{exercise}



\end{document}

