Half Adder, Full Adder, and Subtractors: Design and Working

What are Adders and Subtractors? (In Simple Words)

Inside the CPU, addition is the foundation of all calculations.

• If a computer can add, it can also subtract (using 2’s complement).
• It can multiply (by repeated addition).
• It can divide (by repeated subtraction).

The electronic circuits inside the CPU’s ALU (Arithmetic Logic Unit) that perform this math are called Adders and Subtractors.

💡 Real-Life Indian Student Analogy:
When you add numbers by hand on paper (like 28 + 15):
First you add 8 + 5 = 13. You write down 3 (Sum) and carry over 1 (Carry).
Then in the next column, you must add 2 + 1 plus that carried-over 1!
A Half Adder can only add two numbers. A Full Adder can also include that carried-over number!

1. Half Adder

A Half Adder is a combinational circuit that adds two single-bit binary numbers (A and B).

• It produces two outputs: Sum (S) and Carry (C).

Truth Table:

Boolean Expressions:

• Sum (S): S = A'B + AB' = A ⊕ B (Built using an XOR Gate)
• Carry (C): C = A • B (Built using an AND Gate)

⚠️ Big Limitation of Half Adder: It cannot accept a carry coming from a previous column. Because of this, it can only add the very first column (LSB) of a binary number.

2. Full Adder

A Full Adder fixes the limitation of the Half Adder.

• It adds three 1-bit binary inputs: two main bits (A, B) and a previous carry bit (Cin).
• It produces two outputs: Sum (S) and Carry Out (Cout).

Truth Table:

Boolean Expressions:

• Sum (S): S = A ⊕ B ⊕ Cin
• Carry Out (Cout): Cout = AB + Cin(A ⊕ B) (or AB + BCin + ACin)

Building a Full Adder Using Two Half Adders:

This is a favorite question in DAVV exams! You can build one Full Adder using 2 Half Adders and 1 OR gate:

1. Half Adder 1: Takes inputs A and B. It gives S₁ = A ⊕ B and C₁ = AB.
2. Half Adder 2: Takes S₁ and Cin. It produces the final Sum: S = S₁ ⊕ Cin = A ⊕ B ⊕ Cin and a second carry C₂ = S₁ • Cin.
3. OR Gate: Combines both carries to give final carry: Cout = C₁ + C₂.

3. Half Subtractor

A Half Subtractor subtracts one bit from another (A - B).

• Outputs: Difference (D) and Borrow (Bout).

Truth Table:

Boolean Expressions:

• Difference (D): D = A'B + AB' = A ⊕ B
• Borrow (Bout): Bout = A’ • B (Needs a NOT gate on A, followed by an AND gate)

4. Full Subtractor

A Full Subtractor subtracts three 1-bit inputs:

• First number (A), subtracted number (B), and a previous borrow (Bin).
• Outputs: Difference (D) and Borrow Out (Bout).

Truth Table:

Boolean Expressions:

• Difference (D): D = A ⊕ B ⊕ Bin
• Borrow Out (Bout): Bout = A’B + Bin(A ⊕ B)’ (or A'B + A'Bin + BBin)

💡 DAVV Exam Preparation Tips

5-Mark Question: “Design a Half Adder circuit. Give its truth table, Boolean expressions, and logic diagram.”

• Draw the 4-row truth table for inputs A, B and outputs Sum, Carry.
• Write both formulas clearly: S = A ⊕ B and C = AB.
• Draw the logic circuit showing 1 XOR gate and 1 AND gate.

10-Mark Question: “Explain a Full Adder circuit with its truth table and logic expressions. Show clearly how a Full Adder can be constructed using two Half Adders and an OR gate.”

• Draw the complete 8-row truth table.
• Draw the block diagram showing:
  • First Half Adder (takes A, B)
  • Second Half Adder (takes S₁ and Cin)
  • OR Gate (combines carries C₁ and C₂)
• Write the Boolean equations proving S = A ⊕ B ⊕ Cin and Cout = C₁ + C₂.