---
title: "Full subtractor from half subtractors"
subject: "Digital Logic"
chapter: 5
key: "semester_3/digital_logic:full-subtractor-half-subtractors"
source: https://www.bctengineer.com/q/semester_3/digital_logic/full-subtractor-half-subtractors
site: BCT Engineering
---

# Full subtractor from half subtractors

Digital Logic · Chapter 5: Arithmetic Circuits

- Asked in: 2081 Bhadra, 5 marks
- Video: [Full Subtractor Implementation with Half Subtractors: Designing, Circuit, and Process](https://www.youtube.com/watch?v=GGUpjqj0kGA) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/full-subtractor-half-subtractors
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-5

## Question

Design a full subtractor circuit using half subtractors and one OR gate.

## Short answer

A full subtractor uses two half subtractors: first computes A⊕B and A'B; second computes (A⊕B)⊕Bin giving difference D, and (A⊕B)' Bin. An OR gate combines the borrows: Bo = A'B + (A⊕B)' Bin.

## Answer

A **full subtractor** subtracts $B$ and a borrow-in $B_{in}$ from $A$, giving difference $D$ and borrow-out $B_o$. A **half subtractor** subtracts $Y$ from $X$: $D = X \oplus Y$, $B = X'Y$.

#### Truth table and expressions

| A | B | $B_{in}$ | D | $B_o$ |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 |

$$
D = A \oplus B \oplus B_{in}
$$
$$
\begin{aligned} B_o &= A'B'B_{in} + A'BB_{in}' + A'BB_{in} + ABB_{in} \\ &= A'B(B_{in}' + B_{in}) + B_{in}(A'B' + AB) \\ &= A'B + (A \oplus B)'\,B_{in} \end{aligned}
$$

#### Using two half subtractors and one OR gate

- **HS1** with inputs $A$, $B$: $D_1 = A \oplus B$, $B_1 = A'B$.
- **HS2** with inputs $D_1$, $B_{in}$: $D = D_1 \oplus B_{in} = A \oplus B \oplus B_{in}$, $B_2 = D_1'B_{in} = (A \oplus B)'B_{in}$.
- **OR gate:** $B_o = B_1 + B_2 = A'B + (A \oplus B)'B_{in}$, which matches the expression above.

```text
        +-----+  D1   +-----+
A ----->| HS1 |------>| HS2 |----------> D
B ----->|     |  Bin->|     |
        +-----+       +-----+
           | B1          | B2
           +---->[OR]<---+
                   |
                   +-----------------> Bo
```

**Check:** $A=0, B=1, B_{in}=1$: HS1 gives $D_1 = 1$, $B_1 = 1$; HS2 gives $D = 0$, $B_2 = 0$; $B_o = 1$. So $0 - 1 - 1 = -2$, i.e. $D = 0$ with borrow 1. Correct.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How many half subtractors and OR gates build a full subtractor?
   A. Two half subtractors and one OR gate
   B. One HS and two OR
   C. Three HS
   D. Two HS and no OR
   Answer: A. HS1 on A,B; HS2 on D1,Bin; OR the borrows. (Key point: Hardware)

2. What is D?
   A. A'B
   B. A xor B xor Bin
   C. AB + Bin
   D. (A xor B)'Bin
   Answer: B. Cascaded XORs. (Key point: Difference)

3. What is Bo?
   A. A'B (A xor B)Bin
   B. AB + Bin
   C. A'B + (A xor B)'Bin
   D. A xor B
   Answer: C. OR of the two half subtractor borrows. (Key point: Borrow out)

## Related in Digital Logic

- [2-bit fast adder; full subtractor from half subtractors](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Afast-adder-full-subtractor) — Digital Logic, chapter 5
- [Full subtractor using decoder; combinational circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Afull-subtractor-decoder-combinational) — Digital Logic, chapter 4
- [Half subtractor using HDL](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ahalf-subtractor-hdl) — Digital Logic, chapter 2
- [Half adder/subtractor; 1's and 2's complement subtraction](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ahalf-adder-subtractor-complement) — Digital Logic, chapter 5
- [Full subtractor using a 1:4 DEMUX](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Afull-subtractor-1to4-demux) — Digital Logic, chapter 4, asked 6×
- [Full subtractor using a 2×4 decoder](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Afull-subtractor-2x4-decoder) — Digital Logic, chapter 4, asked 4×

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