Logic Design

HW #$3$

Design a four$-$bit combinational circuit $2\text{'}$s complementer. $($The output generates the $2\text{'}$s complement of the input binary number.$)$ Show that the circuit can be constructed with exclusive$-$OR gates. Can you predict what the output functions are for a five$-$bit $2\text{'}$s complementer?

Design a combinational circuit that compares two $4-$bit numbers to check if they are equal. The circuit output is equal to $1$ if the two numbers are equal and $0$ otherwise.

A combinational circuit is specified by the following two Boolean functions:

$F_1(A,B,C)={\displaystyle \underset{?}{\overset{?}{}}}(1,4)$

$F_2(A,B,C)={\displaystyle \underset{?}{\overset{?}{}}}(2,3,5,6,7)$

Implement the circuit with a decoder constructed with NAND gates and NAND or AND gates connected to the decoder outputs. Use a block diagram for the decoder. Minimize the number of inputs in the external gates.

An $81$ multiplexer has inputs $A,B$ and C connected to the selection inputs $S_2,S_1$ and $S_0,$ respectively. The data inputs $I_0$ through $I_7$ are as follows: $I_1=I_2=0$;$I_3=I_7=1$;$I_4=I_5=D$; and $I_0=I_6=D^{\text{'}}.$ Determine the Boolean function that the multiplexer implements.