Design a combinational circuit that generates the two's complement of a $3-$bit binary code by:

a$.$ Deriving the truth table

b$.$ Determining the output function

c$.$ List essential and non$-$essential prime implicants

d$.$ Drawing the logic diagram

Implement the following combinational circuit with two outputs, $F_1$ and $F_2,$ using a decoder:

$F_1=AC+A^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$

$F_2=A^{\text{'}}B+A{C}^{\text{'}}$

Complete the following truth table. What is this the truth table for? How does it work?

$\backslash$table$[[$Imputs$,$Outputs$],[D_0,D_1,D_2,D_3,D_4,D_5,D_6,D_7,$A$,B,\frac{?}{\text{b}\text{a}\text{r}}(C)$