Answer the following questions, showing your work to receive credit.

Answer the following questions, showing your work to receive credit.

Given $P(w,x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}(0,1,3,4,5,8,10,13,15)$

$($a$)$ Implement a most simplified gate implementation of $P^{\text{'}}$ using only Nand gates. $(10$ points$)$

$($b$)$ Implement $P$ using a minimal number of only Nor gates. $(10$ points$)$

Expand this Boolean Algebraic expression to a fully expanded Sum of Products $($SOP$)$ and identify the minterms. $(15$ points$)$

$T(w,x,y,z)=x^{\text{'}}*z*(w+y^{\text{'}})*(y^{\text{'}}+z)+y*z^{\text{'}}$

Design a digital circuit that compares two $6-$bit values and displays $\text{'}>\text{'},\text{'}=\text{'},$ or $\text{'}<mi\; id="EditorElement\_1200345">\text{'}</mi>$ on a $7-$segment display if $a>b,a=b,$ or $a_5{a}_4{a}_3{a}_2{a}_1{a}_0{b}_5{b}_4{b}_3{b}_2{b}_1{b}_0$a respectively. You may use logic gates and $HA$ and $FA$ blocks $(label$ outputs $on$ each block$)as$ necessary. The input values should $be$ described $by$ these variable names: $a_5{a}_4{a}_3{a}_2{a}_1{a}_0$ and $b_5{b}_4{b}_3{b}_2{b}_1{b}_0.(15$ points$)$

Show a gate implementation $ofa4$ input Decoder. $(15$ points$)$

Implement a circuit $to$ compute the FEC bits for $a7-$bit ASCII encoding $.(20$ points$)$

Design a digital circuit, using logic gates $to$ display the following characters $ona7-$segment display: $P,L,U,S.(15$ points$)$