Answer the following questions $($a$)$ to $($e$).$ In cases where inverted inputs are required as input to the circuit, always use inverted inputs $($e$.$g$.,$ the inverted input of A is denoted as A$’),$ and do not use inverters.

$($a$)$Construct the logic expression F$($A$,$B$,$C$,$D$)=$ A$’$C$+$B$’$C$’+$CD using three two$-$input AND gates and one three$-$input OR gate, and draw the circuit diagram. $$

$($b$)$Construct the circuit from $($a$)$ using three two$-$input NAND gates and one three$-$input NAND gate, and draw the circuit diagram.$$

$($c$)$In the circuit from $($b$),$ a static $1-$hazard occurs. List all the $“$input signal transition patterns$”$ in which a static $1-$hazard occurs. Also, explain the reason for the hazard using a Karnaugh map. Note that the explanation must include the terms $“$gate delay$”$ and $“$input to the NAND gate In the output stage of F$”).$

$($d$)$Draw the circuit diagram for the circuit from $($b$)$ with the minimum number of additional two$-$input NAND gates to prevent static $1-$ hazards. Here, use one three$-$input NAND gate and the minimum number of two$-$input NAND gates. Explain what needs to be added and provide the logic expressions up to the circuit configuration. $$

$($e$)$In the circuit from $($d$),$ when the input $($A$,$B$,$C$,$D$)$ changes from $(1,0,0,1)$ to $(1,1,1,1),$ the output remains one. However, there is a possibility of a hazard occurring in practice. Explain what kind of hazard may occur, referring to the Karnaugh map. Answer $($Answer$1)$~$($Answer$4)$

When the input $($A$,$B$,$C$,$D$)$ transitions from $(1,0,0,1)$ to $(1,1,1,1),$ two inputs need to change, but it is physically impossible for two inputs to change simultaneously. In other words, the inputs change one by one, so either $($A$,$B$,$C$,$D$)($Answer$1)$ or $($Answer$2)$ must be traversed on the Karnaugh map. In the former case, F for $($A$,$B$,$C$,$D$)=($Answer$3)$ is $($Answer$4),$ resulting in a hazard. On the other hand, in the latter case, the circuit has been modified in $($d$)$ to avoid hazards.