Simplify the following Proposition $[10$ marks$]$:

$($a$)(PvvQ{)}^{{?}^{?}}(notPvvnotQ)=(P^{{?}^{?}}$notP${)}^{F}vv(\frac{?}{\text{b}\text{a}\text{r}}(Q{)}^{{?}^{?}}$notQ$)=FvvF=F$

$($b$)$

Draw the logic circuit for the Boolean function

Marks$]$

Logical Equivalence, Conjunction, Disjunction, and Negation $[25$ Marks$]$

Done $1.$ Prove the following logical equivalences $[8$ Marks$]$:

$($a$)Pvv(P^{{?}^{?}}Q)-=P(PvvP{)}^{{?}^{?}}Q=P^{{?}^{?}}Q=P^{{?}^{?}}T-=P$

$($b$)$ not$(P^{{?}^{?}}$notQ$)-=$notPvvQ De Horgan Law $($negate$,$ fliptperation$)$

b$)$