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\textbf{Fuzzy negations, complements, and intersections}

\bigskip
\begin{exercise}\noindent\par
Fuzzy sets $A,B$ have the following vertical representations:
\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}
\\
\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*}

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

\item
Find a fuzzy disjunction dual to the product conjunction with respect to the negation from part~1.
\item
Find the fuzzy complement of $A$ with respect to the negation $\notf\lambda$ from part~1.
\item
Find the standard, product, and \L ukasiewicz intersections of $A$ and~$B$.
\end{enumerate}
\end{exercise}

\end{document}
