Adders

Half adder: $S = A \oplus B$, $C = A \cdot B$.

Full adder (with carry-in):

$S = A \oplus B \oplus C_{in} \qquad C_{out} = AB + C_{in}(A \oplus B)$

Chain $n$ full adders to add $n$-bit numbers (ripple-carry). It's slow because the carry ripples through every stage; carry-lookahead adders compute carries in parallel. Subtraction uses the same adder: $A - B = A + \bar{B} + 1$.