A 2-bit full adder circuit may be designed with the help of the following 2-input logic gates:

Question

Accepted Answer

The correct option is (1):
A full adder is a vital building block in digital electronics, specifically designed to perform the addition of three single-bit inputs: two data bits (A, B) and a carry-in ($C\_{in}$) from a preceding stage. The most straightforward and standard implementation of this circuit is achieved by combining two half adders and one OR gate. The underlying Boolean equations for the full adder outputs—Sum ($S = A \oplus B \oplus C\_{in}$) and Carry-out ($C\_{out} = AB + C\_{in}(A \oplus B)$)—demonstrate the need for five gates. Specifically, two XOR gates are used to compute the two XOR operations for the sum, two AND gates are used to calculate the two product terms that contribute to the carry-out, and one OR gate is used to combine these two carry-out terms, resulting in the correct total count of **two XOR, two AND, and one OR** gates.

The incorrect option is (2):
Option 2, "2-NOR, 2-NAND," is incorrect because this combination and count of gates is insufficient to implement the complex logic of a full adder. While both NAND and NOR gates are classified as universal gates, meaning any logic circuit *can* be built using only one of them, the minimum number of universal gates required for a full adder circuit is significantly higher than four. Specifically, a full adder requires a minimum of nine two-input NAND gates or nine two-input NOR gates for a pure universal gate implementation. Therefore, two NOR gates and two NAND gates are not enough to realize the required sum ($A \oplus B \oplus C\_{in}$) and carry-out logic.

The incorrect option is (3):
Option 3, "1-XOR, 1-AND, 1-OR," describes the component gates necessary to construct a **Half Adder**, not a Full Adder. A half adder only accepts two binary inputs (A and B) and produces a sum and a carry, but it critically lacks the input port for a carry-in bit. Since a full adder is designed precisely to handle this third carry-in input to enable cascading for multi-bit addition, the logic must be more complex, requiring two cascaded half-adder structures along with the final carry-combining OR gate. Thus, the components of a half adder are inadequate for a full adder's function.

The incorrect option is (4):
Option 4, "2-AND, 2-OR," is incorrect because it entirely omits the fundamental **XOR gates** that are essential for the summation function. The primary role of the XOR gate in binary addition is to correctly calculate the Sum bit, based on the principle that the sum is '1' only when an odd number of inputs are '1'. Without any XOR gates, the circuit would be unable to generate the proper Sum output. An adder circuit composed only of AND and OR gates would fail to perform the required modulo-2 addition and could not be classified as a functional binary full adder.