Question

A full adder can be built using:

• A. One Half-Adder (HA)
• B. Two HAs + OR
• C. XOR only
• D. Three HAs

Hint:

To build a Full Adder, you need exactly 2 Half Adders and 1 OR gate.
In terms of basic gates, a Full Adder can be constructed using 2 XOR gates, 2 AND gates, and 1 OR gate.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks about the constituent components required to construct a Full Adder circuit using Half Adder (HA) blocks.
An adder is an essential block in arithmetic units of processors, handling the summation of binary digits.

Step 2: Key Formula or Approach:
The logic equations for a Half Adder with inputs \( A \) and \( B \) are:
\[ S_{\text{HA}} = A \oplus B \] \[ C_{\text{HA}} = AB \] The logic equations for a Full Adder with inputs \( A, B, \) and carry-in \( C_{\text{in}} \) are:
\[ S_{\text{FA}} = A \oplus B \oplus C_{\text{in}} \] \[ C_{\text{out}} = AB + BC_{\text{in}} + C_{\text{in}}A \] By rewriting the carry-out equation, we can map it to Half Adder blocks:
\[ C_{\text{out}} = AB + C_{\text{in}}(A \oplus B) \]

Step 3: Detailed Explanation:
Let us physically trace how two Half Adders and an OR gate are connected to form a Full Adder:

• First Half Adder (HA 1):
Inputs connected: \( A \) and \( B \).
Outputs obtained:
Sum 1: \( S_1 = A \oplus B \)
Carry 1: \( C_1 = AB \)

• Second Half Adder (HA 2):
Inputs connected: \( S_1 \) (from HA 1) and \( C_{\text{in}} \).
Outputs obtained:
Sum 2 (Final Sum): \( S_{\text{FA}} = S_1 \oplus C_{\text{in}} = (A \oplus B) \oplus C_{\text{in}} \)
Carry 2: \( C_2 = S_1 \cdot C_{\text{in}} = (A \oplus B)C_{\text{in}} \)

• OR Gate Combination:
To find the final carry-out, we combine the carry outputs of both Half Adders using a 2-input OR gate:
\[ C_{\text{out}} = C_1 + C_2 = AB + (A \oplus B)C_{\text{in}} \] This combination successfully implements both the Sum and Carry equations of a Full Adder.

Step 4: Final Answer:
A Full Adder is built using two Half-Adders (HAs) and one OR gate.