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$($a$)$ Given the linguistic input variables, Age and Height, each with the given terms:

Age

young $=\frac{1}{0}+\frac{0\mathrm{.}9}{10}+\frac{0\mathrm{.}8}{20}+\frac{0\mathrm{.}5}{30}+\frac{0\mathrm{.}2}{40}+\frac{0\mathrm{.}0}{50}$

middle $=\frac{0\mathrm{.}2}{20}+\frac{0\mathrm{.}5}{30}+\frac{0\mathrm{.}7}{40}+\frac{1}{50}+\frac{0\mathrm{.}7}{50}+\frac{0\mathrm{.}5}{60}+\frac{0\mathrm{.}2}{70}$

old $=\frac{0\mathrm{.}1}{40}+\frac{0\mathrm{.}3}{50}+\frac{0\mathrm{.}5}{60}+\frac{0\mathrm{.}7}{70}+\frac{0\mathrm{.}9}{80}+\frac{1}{90}+\frac{1}{100}$

Height

short $=\frac{1}{1\mathrm{.}4}+\frac{0\mathrm{.}75}{1\mathrm{.}5}+\frac{0\mathrm{.}5}{1\mathrm{.}6}+\frac{0\mathrm{.}25}{1\mathrm{.}7}$

average $=\frac{0\mathrm{.}5}{1\mathrm{.}6}+\frac{1}{0\mathrm{.}7}+\frac{0\mathrm{.}5}{1\mathrm{.}8}$

tall $=\frac{0\mathrm{.}2}{1\mathrm{.}6}+\frac{0\mathrm{.}5}{1\mathrm{.}7}+\frac{0\mathrm{.}75}{1\mathrm{.}8}+\frac{1}{1\mathrm{.}9}+\frac{1}{2\mathrm{.}0}$

and the single linguistic output variable Employ, with the three terms:

Employ

bad $=\frac{1}{0}+\frac{1}{1}+\frac{0\mathrm{.}7}{2}+\frac{0\mathrm{.}5}{3}+\frac{0\mathrm{.}3}{4}$

fair $=\frac{0\mathrm{.}1}{3}+\frac{0\mathrm{.}5}{4}+\frac{1}{5}+\frac{0\mathrm{.}5}{6}+\frac{0\mathrm{.}1}{7}$

good $=\frac{0\mathrm{.}3}{6}+\frac{0\mathrm{.}5}{7}+\frac{0\mathrm{.}7}{8}+\frac{1}{9}+\frac{1}{10}$

calculate the fuzzy set result of applying Mamdani inference on input values of Age$=40$ and Height $=1\mathrm{.}6$ when using the rules:

R$1$: IF Age is young OR Height is tall THEN Employ is good

R$2$: IF Age is middle THEN Employ is fair

R$3$: IF Age is old AND Height is short THEN Employ is bad

Note that you should not defuzzify the output set at this stage.

$($b$)$ Describe and give basic formulae for two alternative methods of numeric defuzzification.

Provide an example of two obviously different output sets for which one of these defuzzification methods gives the same result.

$($c$)$ Calculate the defuzzified value of your output set from Q$2($a$)$ above, using centre$-$of$-$gravity defuzzification. If you have not been able to derive a set for Q$2($a$),$ then write down your own fuzzy set defined on a universe of discourse of the integers from $0$ to $10,$ and calculate its defuzzified value.

$($d$)$ Describe the role of similarity measures in the process of linguistic defuzzification. Give examples of two different similarity measures.

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