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1. Introduction

In Computer System Architecture (CSA), half adder and half subtractor are basic combinational logic circuits used to perform simple binary addition and binary subtraction of two single-bit inputs.

These are fundamental building blocks for designing complex arithmetic operations inside processors (ALU – Arithmetic Logic Units).

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2. Half Adder

Definition

A Half Adder is a combinational circuit that performs the addition of two single-bit binary numbers.
It produces two outputs:

• Sum (S): Represents the result of addition.
• Carry (C): Represents overflow if the sum exceeds 1.

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Inputs and Outputs

• Inputs: A, B (two binary digits)
• Outputs: Sum (S), Carry (C)

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Truth Table

A	B	Sum (S)	Carry (C)
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

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Boolean Expressions

• Sum (S) = A ⊕ B (A XOR B)
• Carry (C) = A · B (A AND B)

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Logic Diagram

  A -----|⊕|---- Sum
          |
          |         
   B -----|

   A -----|&|---- Carry
          |
          |
   B -----|

• XOR Gate for Sum
• AND Gate for Carry

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Working

• If both inputs are 0 → Sum = 0, Carry = 0
• If one input is 1 → Sum = 1, Carry = 0
• If both inputs are 1 → Sum = 0, Carry = 1 (because 1 + 1 = 10 in binary)

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3. Half Subtractor

Definition

A Half Subtractor is a combinational circuit that performs the subtraction of two single-bit binary numbers.
It produces two outputs:

• Difference (D): Represents the result of subtraction.
• Borrow (B): Represents if a ‘borrow’ is needed.

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Inputs and Outputs

• Inputs: A, B (minuend and subtrahend)
• Outputs: Difference (D), Borrow (B)

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Truth Table

A	B	Difference (D)	Borrow (B)
0	0	0	0
0	1	1	1
1	0	1	0
1	1	0	0

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Boolean Expressions

• Difference (D) = A ⊕ B (A XOR B)
• Borrow (B) = (¬A) · B

Here:

• ¬A means NOT A.
• Borrow occurs when A is 0 and B is 1.

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Logic Diagram

  A -----|⊕|---- Difference
          |
          |         
   B -----|

   A ---|¬|--+---|&|---- Borrow
               |
   B -----------|

• XOR Gate for Difference
• NOT gate and AND gate combination for Borrow

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Working

• If both inputs are 0 → Difference = 0, Borrow = 0
• If A is 0 and B is 1 → Difference = 1, Borrow = 1 (because you need to borrow 1)
• If A is 1 and B is 0 → Difference = 1, Borrow = 0
• If both are 1 → Difference = 0, Borrow = 0

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🌟 Summary Table

Feature	Half Adder	Half Subtractor
Purpose	Adds two bits	Subtracts two bits
Inputs	A, B	A, B
Outputs	Sum, Carry	Difference, Borrow
Sum/Diff	A ⊕ B	A ⊕ B
Carry/Borrow	A · B	(¬A) · B

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🎯 Key Points to Remember

• Half Adder: Adds two bits — useful in basic addition units.
• Half Subtractor: Subtracts two bits — useful in arithmetic circuits.
• XOR Gate is used in both circuits for Sum and Difference.
• Half Adder cannot add three bits (for that, we use Full Adder).
• Half Subtractor cannot handle borrow from previous stage (for that, we use Full Subtractor).