Q.What is a full adder? Explain its working with respect to three input X-OR gate and basic gates with the help of truth table and Boolean expression.

A full adder adds three bits — two operand bits and a carry-in — producing a sum and a carry-out, and can be built using a three-input X-OR gate together with AND and OR gates.

[!ANSWER]

A full adder gives $Sum=A\oplus B\oplus C_{in}$ (a three-input X-OR) and $C_{out}=AB+BC_{in}+AC_{in}$ (AND gates ORed together); the sum is HIGH for an odd number of input 1s and the carry-out is HIGH when two or more inputs are 1.

[!TLDR]

Full adder: $Sum=A\oplus B\oplus C_{in}$ via a 3-input X-OR, $C_{out}=AB+BC_{in}+AC_{in}$ via AND-OR logic.

Definition: A full adder is a combinational circuit that adds three input bits — the two operand bits $A$ and $B$ and the carry-in $C_{in}$ from the previous stage — and produces a Sum and a carry-out $C_{out}$. Unlike the half adder, it accounts for the incoming carry, so full adders can be cascaded to add multi-bit numbers.

Truth table:

Sum — three-input X-OR: The Sum column is 1 whenever an odd number of inputs are 1, which is exactly the three-input X-OR (odd) function: $Sum = A\oplus B\oplus C_{in}$. It is realised by cascading two 2-input X-OR gates — the first forms $A\oplus B$, the second X-ORs that with $C_{in}$.

Carry-out — basic gates: The carry-out is 1 whenever any two or more inputs are 1: $C_{out} = AB + BC_{in} + AC_{in}$. This is realised with three AND gates feeding an OR gate; equivalently $C_{out}=AB+(A\oplus B)C_{in}$, using the $A\oplus B$ term already formed for the sum.

Logic diagram: two X-OR gates in cascade give the Sum; AND gates for $AB$, $BC_{in}$, $AC_{in}$ ORed together give the carry-out.

[!ANSWER]

A full adder adds $A$, $B$ and $C_{in}$: the sum is the three-input X-OR $Sum=A\oplus B\oplus C_{in}$ (odd function) and the carry-out is $C_{out}=AB+BC_{in}+AC_{in}$ from AND gates ORed together, as verified by the truth table above.