One of the difficulties with the Gaussian probability distribution is that it has no closed-form integral. Integration of this function must be conducted numerically. Because of this difficulty, approximations to the Gaussian distribution have been developed over the years. One of these approximations is the expression shown in Fig. P8.3a.
This expression provides a reasonably good approximation to the Gaussian except for values of $x$ near zero; as can be seen, the function $G$ has a singularity at $x=0$. Table P8.3 shows the exact values for this approximate function, $G$, and Fig. P8.3a shows the function.
If one uses the partitioning for the input variable, $x$, as shown in Fig. P8.3b, the discrete membership values for each of the quantities $x$ shown in Table P8.3 for the following three fuzzy inputs,
1. $x_1=\mathrm{NB}$ or PB
2. $x_2=\mathrm{Z}$ or PS
3. $x_3=\mathrm{Z}$ or NS
would be
$$
\begin{aligned}
& x_1=[1,0.5,0,0,0,0,0,0.5,1] \\
& x_2=[0,0,0.0 .5,1,0.5,1,0.5,0] \\
& x_3=[0,0.5,1,0.5,1,0.5,0,0.0]
\end{aligned}
$$
The membership functions for $G$ for the first five elements in the table (the function is symmetric) corresponding to the three fuzzy inputs are
$$
\begin{aligned}
G_1 & =\left\{\frac{1}{0}+\frac{0.5}{0.5}+\frac{0}{0.776}+\frac{0}{1.10}+\frac{0}{2}\right\}=[1,0.5,0,0,0] \\
G_2 & =[0,0,0,0.5,1] \\
G_3 & =[0,0.5,1,0.5,1]
\end{aligned}
$$
(a) Develop fuzzy relations (these matrices all will be of size $9 \times 5$ ) between the three fuzzy inputs and outputs using a Cartesian product operation.
(b) Find the overall fuzzy relation by taking the union of the three relations found in part (a).
(c) If the matrix relation in part (b) is replaced by a continuous surface, composition with crisp singleton inputs for $x$ results in the following table of results for the output $G$. Verify some of these results.