Question $2$: Combinational Circuit Design $(30$ points$)$

The water level of a chemical reactor is represented by a $3-$bit unsigned integer number. Therefore, we use $3$ Boolean variables XYV for inputs, with $\backslash ($ X $\backslash )$ as the MSB$.$ In addition, there is an input $\backslash ($ W $\backslash )$ to indicate the criticality of the water level, such that output $\backslash ($ Z$=$I $\backslash )$ when: $($i$)\backslash ($ W$=0\backslash )$ and the water level is NOT between $2\mathrm{.}5.$ or $($ii$)\backslash ($ W$=$I $\backslash )$ and the water level is NOT between $3\mathrm{.}4.$ Finally, there are $4$ output signals ated to represent the input value XIV in BCD code.

Design a combinational circuit to implement the above process, providing:

$($a$)$ The truth table $(10$ points$)$

$($b$)$ A minimum logic diagram using only AND, OR and NOT gates $(10$ points$)$

$($c$)$ An implementation using a $4-$to$-16.$ Decoder and any other necessary gates $(5$ points$)$

$($d$)$ An implementation using Multiplexers and any other necessary gates. State the size and number of the neceessary multiplexers $(5$ points$)$