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\begin{document}

\textbf{Conversions of representations of fuzzy sets, fuzzy negations, fuzzy complements}

\bigskip
\begin{exercise}
Fuzzy set $C$ is given by its collection of cuts:
$$\rezR_C(\alpha)=
\begin{cases}
 \R\,, & \alpha=0\,,\\
(\alpha, 6-2\,\alpha^2)\,, & \alpha\in(0,1]\,.\\
\end{cases}
$$
Find its vertical representation.
\end{exercise}

\begin{exercise}\noindent\par
Find for which $\lambda$ the formula
$$\notf\lambda \alpha =1- \sqrt{1-(1-\alpha)^\lambda}$$
defines a fuzzy negation~$\notf\lambda$.
Draw a graph.
\end{exercise}

\begin{exercise}\noindent\par
Fuzzy set $A$ has the following vertical representation:
\begin{align*}
\mu_A(x)=
\begin{cases}
 \dfrac{x-1}2\,, & x\in [1,3]\,,\\
 1\,, & x\in [3,6]\,,\\
 4-\dfrac x 2\,, & x\in [6,8]\,,\\
0\,, & \text{otherwise}\,.\\
\end{cases}
\end{align*}
Find its fuzzy complement with respect to 
\begin{enumerate}
\item the standard negation,
\item the negation $\notf\lambda$ from the preceding exercise.
\end{enumerate}
\end{exercise}

\end{document}
