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\begin{document}
Exercise from fuzzy intersections.

\begin{exercise}
Fuzzy sets $A,B$ have the following vertical representations:
\begin{align*}
\mu_A(x)=
\begin{cases}
 \dfrac{x-1}3\,, & x\in [1,4]\,,\\
 1\,, & x\in [4,6]\,,\\
 7-x\,, & x\in [6,7]\,,\\
0\,, & \text{otherwise}\,,\\
\end{cases}
\\
\mu_B(x)=
\begin{cases}
 \dfrac{x-2}3\,, & x\in [2,5]\,,\\[.5em]
 1\,, & x\in [5,6]\,,\\
 \dfrac{8-x} 2\,, & x\in [6,8]\,,\\
0\,, & \text{otherwise}\,.\\
\end{cases}
\end{align*}
Find their standard, product, and \L ukasiewicz intersections.
\end{exercise}

Exercise from fuzzy algebras.

\begin{exercise}
Decide whether the equality
$$\alpha\orf.(\alpha\andf. \beta)=\alpha$$
holds for all $\alpha, \beta$ for the following choices of operations $\andf.,\orf.$:

\begin{enumerate}
\item $\andf S, \orf S$,
\item $\andf P, \orf S$,
\item $\andf P, \orf L$,
\item $\andf L, \orf P$,
\item $\andf P, \orf S$,
\item $\andf L, \orf L$.
\end{enumerate}
\end{exercise}

\end{document}

