Building a full adder from half adders: Evaluate the claim: “Two half adders can be combined to form a full adder with no additional gates.” State whether the statement is valid.

Introduction / Context:
Adder construction is a staple topic in digital design. A full adder sums three one-bit inputs (A, B, Cin) to produce Sum and Cout. A common synthesis is to use two half adders plus an OR gate. This question tests whether you recall the exact gate count and interconnection required.

Given Data / Assumptions:

• Half adder produces Sum = A XOR B and Carry = A AND B.
• Full adder requires Sum = A XOR B XOR Cin and Cout = majority(A,B,Cin).
• Standard gates (no wired-logic assumptions).

Concept / Approach:
Construct a full adder by: (1) Half-adder #1 on A and B → S1 = A XOR B, C1 = A AND B. (2) Half-adder #2 on S1 and Cin → Sum = S1 XOR Cin, C2 = S1 AND Cin. (3) Final carry Cout = C1 OR C2. Therefore, two half adders alone do not complete the full adder; an additional OR gate is required to combine the two carry terms.

Step-by-Step Solution:

Verification / Alternative check:
Truth-table verification of carry terms shows Cout true whenever at least two of A, B, Cin are 1. C1 OR C2 realizes this, requiring the final OR gate beyond the two half adders.

Why Other Options Are Wrong:

• Correct: Omits the necessary OR gate.
• Wired-OR/XOR–XNOR sharing/fan-out only: These are implementation details that do not remove the logical need to combine carry terms.

Common Pitfalls:
Forgetting the final carry combination stage; assuming the second half adder magically produces the final Cout directly without merging intermediate carries.

Final Answer:
Incorrect