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\begin{document}
Exercises from conversions from horizontal to vertical representation of fuzzy sets and fuzzy negations.

%\begin{enumerate}
%\item 
\begin{exercise}[``satisfaction with the grades''; old scale]
Fuzzy set $A$ is given by its collection of cuts:
$$\rezR_A(\alpha)=
\begin{cases}
 \{1,2,3,4\}\,, & \alpha=0,\\
 \{1,2\}\,, & \alpha\in(0,0.5],\\
 \{1,2,3\}\,, & \alpha\in(0.5,0.8],\\
 \{1\}\,, & \alpha\in(0.7,1].
\end{cases}
$$
Find its vertical representation.
\end{exercise}

%\item 
\begin{exercise}
Verify that $\notf.$ is a fuzzy negation and $i_1,i_2$ are its different generators:
\begin{align*}
\notf.\alpha &= \frac{1-\alpha} {1+\alpha} \,,\\
i_1(\alpha) &= \log_2(1+\alpha) \,,\\
i_2(\alpha) &= \frac{3\alpha+\alpha^2} {2(1+\alpha)} \,.
\end{align*}\end{exercise}
%\end{enumerate}

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