$($a$)$ Write the Boolean function & truth table that resembles the circuit shown in Fig. $2$ Please show your calculations and don't provide only final answers

Question $1$:

$[(x+x^{\text{'}}*y+x^{\text{'}}*y^{\text{'}})*(x+x*y+x*y^{\text{'}}){]}^{\text{'}}+x*y^{\text{'}}+yz$

$($a$)$ Simplify the Boolean function $($i$).$

$(2$ points$)$

$($b$)$ Write the truth table of the Boolean function of question $1($a$)$

$($c$)$ Draw the simplified Boolean function in question $1($a$)$ using OR & NOT gate.

$($d$)$ Draw the simplified Boolean function in question $1($a$)$ using AND & NOT gate.

Question $2$:

Figure $1$

$($a$)$ Write the Boolean function that resembles the circuit shown in Figure $1$

$($b$)$ Simplify the Boolean function of question $2($a$)$

$($c$)$ Draw the simplified Boolean function of question $2($b$)$ using AND, OR$,$ and NOT gates.

Question $3$: Sum of Product

$($a$)$ Write down and simplify the following Boolean function $f(x1,x2,x3)={\displaystyle \underset{?}{\overset{?}{}}}(1,2,3,4,5,6,7)$

$($b$)$ Write down and simplify the following Boolean function $f(x1,x2,x3,x4)=$

$(1,3,5,7,13,15)$

$($b$)$ Lets assume that the AND gate is faulty and this means $90\%$ of the times it will give the right

output and $10\%$ of the time it will give the wrong output and so the truth table will look like the

following

Draw the truth table of the entire circuit using the faulty AND gate and comment if this will

affect the output of the entire circuit $(f).$

$($c$)$ Consider a scenario where you have the ability to modify the behavior of the malfunctioning AND

gate shown above. Currently, this gate functions correctly as an AND gate $90\%$ of the time, but

malfunctions $10\%$ of the time, acting as a NAND gate instead. Is it possible for this $10\%$

malfunctioning behavior to mimic a different logic gate $($other than an AND gate$)$ while still

producing the correct output? Please explore alternative configurations or gate types that could

replace the NAND gate behavior during the malfunctioning period, ensuring the overall system's

output remains accurate.