$15$ points$]$ Assume that you are a CPEG student and you are hired as an

intern by Arman for his hardware company that only utilizes NOR $()$

gates for logic operations. NOR gate is implemented as p$$q $=($p $$ q$).$

Recall that the OR gate for inputs p$,$ q is calculated as p $$ q$.$ NOT gate

of p is calculated as $$p and the AND gate of inputs p$,$ q is calculated as

p $$ q$.$

$($a$)[6$ points$]$At the onboarding training, your boss Arman said that

NOR gate is a universal gate and it is possible to create AND, OR$,$

and NOT gates by solely using NOR gates. Please use what you

have learned in COMP$2711$ to prove it by providing the equivalent

representations of AND, OR$,$ and NOT gates using only NOR gates.

$($b$)[9$ points$]$ Because of your excellent performance during training, Ar$-$

man assigns you a real project. In this project, you need to design

the logical expression $(($q $$ r$)->$ p$)$ using only NOR $()$ gates.

What will be the equivalent expression?

Hint: there are several ways to convert this. One approach is to

apply what you have studied on top and get a very large statement.

However there exists an elegant and short way to express this state$-$

ment using NOR gates only. The idea is to somehow get statements

like $($x $$ y$)$ so that you can immediately use x$$y instead