$(10$ points$)$ Simplify the following functions using K$-$Maps to the minimum POSs:$(25$ points$)$ Consider the problem of designing a combinational logic circuit that converts the first eleven

binary numbers to a customized Gray code as shown in the table below. The unused binary combinations

should be used as don't$-$care conditions.

In this question, you are required to do the following:

a$.$ Build the truth table of the combination logic circuit. The inputs $($the four bits of the binary code$)$

of the truth table should be called $(\{$:$x_3{x}_2{x}_1{x}_0)$ and should be sorted in same order as shown in the

table above. Also, the outputs $($the four bits of the gray code$)$ of the truth table should be called

$(x_3{x}_2{x}_1{x}_0).$ You need to use K$-$maps to simplify the outputs.

$($a$)F(x,y,z,w)=(1,3,5,7,13,15)$

$($b$)F(x,y,z,w)={\displaystyle \underset{m}{\overset{?}{}}}(0,6,8,13,14)+{\displaystyle \underset{d}{\overset{?}{}}}(2,4,10)$