Question $3$: Design the circuitry for the "move" function for a three$-$floor elevator. Two$-$bit signal $(F_1{F}_0)$ represents the desired destination: $01=$ first floor, $10=$ second floor, and $11=$ third floor. The present floor indicator requires two bits $(x$ and $y)$: $01$ for first floor, $10$ for second floor, and $11$ for third floor. $($Note that $00$ is an impossible combination and should be omitted from the table.$)$ For the output $M_1{M}_0$:$01$ means moving up; $10$ means moving down; $00$ means stay.$|$

a$.$ Construct the truth table; optimize output Boolean functions and draw the circuits using logic gates.

b$.$ Redraw the circuit for M$1$ using NAND gates only.