Part $1$: Full adder circuit

Full Adder is the adder that adds three inputs and produces two outputs. The first two inputs are

A and B and the third input is an input carry as C$-$IN$.$ The output carry is designated as C$-$OUT

and the normal output is designated as S which is SUM.

we use a full adder because when a carry$-$in bit is available, another $1-$bit adder must be used

since a $1-$bit half$-$adder does not take a carry$-$in bit. A $1-$bit full adder adds three operands and

generates $2-$bit results