fulladder

A full adder is a digital logic circuit that adds three one-bit inputs: A, B, and carry-in (Cin). It produces two outputs: a sum bit (S) and a carry-out bit (Cout). It is the standard building block for adding binary numbers in computers and other digital systems.

The sum is the XOR of the three inputs: S = A XOR B XOR Cin. The carry-out is true when at least two of the inputs are 1, expressed as Cout = (A AND B) OR (B AND Cin) OR (A AND Cin). An equivalent form is Cout = (A AND B) OR (Cin AND (A XOR B)).

A common implementation uses two cascaded half adders. The first half adder adds A and B to produce S1 and C1. The second half adder adds S1 and Cin to yield the final Sum and C2. The overall carry-out is Cout = C1 OR C2. Other gate-level designs exist, but they realize the same Boolean functions for S and Cout.

In practical circuits, full adders are used to build multi-bit adders such as ripple-carry adders, where the carry-out of one stage feeds the carry-in of the next. They are fundamental in arithmetic logic units (ALUs) and various arithmetic circuits. Realizations vary with technology (for example CMOS or TTL) and may optimize for speed, area, or power, but the basic three-input, two-output behavior remains the same.

See also: half adder, ripple-carry adder, binary adder.