Part $1$: Combinational Circuits $(9$ points$)$

The following table shows a $4-$to$-2$ priority encoder with inputs $A,B,C,D$ and outputs $P$ and $Q.\text{'}x\text{'}$ represents don't$-$care value. The simplified SOP expressions for $P(A,B,C,D)$ is:

A$.A^{\text{'}}*+B^{\text{'}}*C^{\text{'}}*D$

B$.D+B^{\text{'}}*C$

C$.B+D$

$\backslash$table$[[A,B,C,D,P,Q$