48. Full Adder Design

• Output Sum Analysis: The Sum output turns on (1) whenever an odd number of inputs are high (either exactly one input is 1, or all three are 1). This is built using two XOR gates connected in a row.
• Output Carry Analysis: The Carry output turns on (1) whenever any two or three inputs are high. To save space and use fewer gates, the circuit reuses the calculation from the very first XOR gate instead of building a whole new path from scratch.
• Hardware Optimization: Input A and Input B go into a first set of gates to get a temporary sum and a temporary carry. These temporary results are then combined with Carry In using shared parts, keeping the total gate count as small as possible.
• Boolean Equations:

Output Sum = (Input A ^ Input B) ^ Carry In

Output Carry = (Input A . Input B) + (Carry In . (Input A ^ Input B))