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\textit{Homework from Fuzzy Logic.}

\bigskip
Exercises from conversions of representations of continuous fuzzy sets.

\begin{exercise}
Fuzzy set $A$ has the following vertical representation:
$$\mu_A(x)=
\begin{cases}
 1\,, & x\in [1,2]\,,\\
 \dfrac{x+2}3\,, & x\in [-2,1)\,,\\[.5em]
\dfrac{4-x}2\,, & x\in (2,4]\,,\\
0\,, & \text{otherwise}\,.\\
\end{cases}
$$
Find the horizontal representation of this fuzzy set.
\end{exercise}

\begin{exercise}
Fuzzy set $B$ is given by its collection of cuts:
$$\rezR_B(\alpha)=
\begin{cases}
 \R\,, & \alpha=0\,,\\
[3\,\alpha, 17-2\,\alpha]\,, & \alpha\in(0,1]\,.\\
\end{cases}
$$
Find its vertical representation.
\end{exercise}

\begin{exercise}
Fuzzy set $C$ is given by its collection of cuts:
$$\rezR_C(\alpha)=
\begin{cases}
 \R\,, & \alpha=0\,,\\
(3\,\alpha, 17-2\,\alpha]\,, & \alpha\in(0,1]\,.\\
\end{cases}
$$
Find its vertical representation.
\end{exercise}

\begin{exercise}
Fuzzy set $D$ is given by its collection of cuts:
$$\rezR_D(\alpha)=
\begin{cases}
 \R\,, & \alpha=0\,,\\
[3\,\alpha, 17-2\,\alpha]\,, & \alpha\in(0,1)\,,\\
\{4\}\,, & \alpha=1\,.\\
\end{cases}
$$
Find its vertical representation.
\end{exercise}

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