In the field of image processing, a limiter function is used to enhance an image

when background lighting is too high. The limiter function is shown in Figure P$8\mathrm{.}7$a$.$

$($a$)$ Using the following rules, construct three matrix relations using the input

$($Figure P$8\mathrm{.}7$b$)$ and output $($Figure P$8\mathrm{.}7$c$)$ partitions:

Rule $1$: IF x $=$ Z$,$ THEN y $=$ S

Rule $2$: IF x $=$ PB$,$ THEN y $=$ PM

Rule $3$: IF x $=$ NB$,$ THEN y $=$ NM$.$