Please do by hand only. draw diagram by hand only! and do all steps both problems Design a Single cell $-1$ bit Carry propagate $($Ripple Carry Adder$)$ full adder.

a$.$ Generate the truth table

b$.$ Using K$-$map, determine the logical expression for Carry out $($C$-$out$)$ and Sum $($S$)$

c$.$ Based on the logical expression, create the schematic diagram for full adder Design a $1$ bit, $2$ to $1$ multiplexer $($Mux$).$ Outputs Y when S $=0$; X when S $=1.$

a$.$ Generate the truth table

b$.$ Using K$-$map, determine the logical expression for output

c$.$ Based on the logical expression, create the schematic diagram for Mux