Consider a simple case presented in the following figure, where there are two input variables $x$

and $Y$ and one output variable $Z.$ Let us also assume the following fuzzy rules:

IF $x$ is low and $Y$ is low THEN $Z$ is high

IF $x$ is high and $Y$ is high THEN $Z$ is low

For $x=55$ and $Y=100$ :

a$)$ Determine the fuzzy outputs sets produced by the min and prod inferencing for $Z$

b$)$ Determine the combined fuzzy output sets by max and sum compositions for $Z.$

c$)$ Find the crisp output values of $Z$ from combined output sets using centroid $($COG$),$ BOA,

MM $($maximum membership$),$ weighted average, LMM and RMM$,$ and average of

maxima defuzzification methods.