Adder

A Boolean circuit computing the sum of binary representations, possibly including input and output carry bits. Its delay depends on how carries are propagated; specifying an adder does not by itself imply constant circuit depth as the input width grows.

A combinational logic circuit adding two input bits $a,b$ to a sum bit $s=a\oplus b$ and carry bit $c=a\wedge b$. An exclusive or gate and an AND gate implement it. The identity $a+b=(a\oplus b)+2(a\wedge b)$ follows by checking the four input pairs. A half adder has no input carry, so adding three equally weighted bits needs a different circuit or multiple stages.