Half Subtractor

Definition

A half subtractor is a combinational logic circuit that subtracts one binary digit from another and generates two outputs:

Difference (D)

• : the result of the subtraction

Borrow (B)

• : an output indicating whether a borrow is needed from the next higher bit

It accepts two input bits:

Minuend (A)

• : the number from which another number is subtracted

Subtrahend (B)

• : the number being subtracted

The half subtractor performs the operation:

Difference = A ⊕ B

Borrow = A' · B

where:

⊕

• denotes XOR

'

• denotes NOT (complement)

·

• denotes AND

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Main Content

1. Inputs and Outputs of Half Subtractor

• The half subtractor has two binary inputs, usually named A and B, where:
• A is the minuend
• B is the subtrahend
• It produces two outputs:
• Difference (D), which is the binary result after subtraction
• Borrow (Borrow out), which indicates whether subtraction required borrowing from the next higher bit
• Since it works only on one-bit subtraction, it cannot accept a previous borrow input. This is the main limitation that distinguishes it from a full subtractor.

Truth table of half subtractor

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

Explanation of each case:

0 − 0 = 0

• , no borrow

0 − 1 = 1

• with borrow, because 0 cannot subtract 1 directly

1 − 0 = 1

• , no borrow

1 − 1 = 0

• , no borrow

This truth table is the foundation for deriving the logic expressions and implementing the circuit.

2. Logic Expressions and Boolean Derivation

• The difference output is obtained using the XOR gate:
• D = A ⊕ B
• XOR gives 1 only when the inputs are different, which matches subtraction behavior for one-bit values
• The borrow output is obtained using NOT and AND:
• Borrow = A' · B
• Borrow occurs only when A = 0 and B = 1, meaning the minuend is smaller than the subtrahend
• These expressions can be derived directly from the truth table using Boolean algebra or Karnaugh maps

Why XOR for Difference?

• If both bits are equal, the result is 0
• If they are different, the result is 1
• This exactly matches the subtraction result for 1-bit binary arithmetic

Why A'·B for Borrow?

• Borrow is needed only when:
• the minuend bit A is 0
• the subtrahend bit B is 1
• That condition is represented by A' AND B

This makes the half subtractor very simple and efficient in hardware design.

3. Circuit Implementation and Symbolic Representation

• A half subtractor can be implemented using:
• 1 XOR gate for Difference
• 1 NOT gate and 1 AND gate for Borrow
• The logic circuit is compact and low-cost, making it useful in basic arithmetic designs

Functional arrangement:

A ─────┬───────────┐
       │           │
       │         ┌─▼─┐
       │         │XOR│─── Difference
       │         └─▲─┘
       │           │
B ─────┼───────────┘

A ──► NOT ──┐
            ├──► AND ─── Borrow
B ──────────┘

Interpretation of the circuit:

• The XOR gate compares the two bits and produces the difference
• The NOT gate inverts A
• The AND gate combines A' and B to produce the borrow

Symbolic view:

• Input: A, B
• Output: D, Borrow

This simple structure makes the half subtractor an important introductory circuit in digital logic design.

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Working / Process

1. Apply the two binary inputs

• Give the minuend bit A and subtrahend bit B to the circuit
• The circuit evaluates both inputs simultaneously because it is a combinational circuit

2. Generate the difference output

• The XOR gate checks whether the input bits are the same or different
• If they are the same, the result is 0
• If they are different, the result is 1
• This becomes the Difference output

3. Generate the borrow output

• The NOT gate first complements A
• The AND gate then checks whether A is 0 and B is 1
• If that condition is true, the circuit outputs Borrow = 1
• Otherwise, borrow remains 0

Example 1: A = 0, B = 1

• Difference = 0 ⊕ 1 = 1
• Borrow = 0' · 1 = 1
• Final output: D = 1, Borrow = 1

Example 2: A = 1, B = 0

• Difference = 1 ⊕ 0 = 1
• Borrow = 1' · 0 = 0
• Final output: D = 1, Borrow = 0

This process shows how the half subtractor performs subtraction for one-bit binary numbers instantly through logic gates.

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Advantages / Applications

Simple and easy to implement

• The half subtractor uses only a few basic logic gates, so it is easy to design and understand

Useful for learning binary arithmetic

• It helps students understand the relationship between binary subtraction, borrow, and logic gate operations

Foundation for larger subtractor circuits

• Half subtractors are used as building blocks in the construction of full subtractors and multi-bit binary subtraction circuits

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Summary

• A half subtractor is a combinational circuit that subtracts one binary bit from another
• It produces two outputs: difference and borrow
• The difference is generated using XOR, and the borrow is generated using NOT and AND

Important terms to remember

• Minuend
• Subtrahend
• Difference
• Borrow
• XOR gate
• AND gate
• NOT gate