---
title: "2's complement subtraction; AND using NOR"
subject: "Basic Electronics Engineering"
chapter: 6
key: "semester_2/basic_electronics_engineering:two-complement-111-110-nor-and"
source: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/two-complement-111-110-nor-and
site: BCT Engineering
---

# 2's complement subtraction; AND using NOR

Basic Electronics Engineering · Chapter 6: Digital Electronics

- Asked in: 2068 Bhadra, 3+3 marks
- Video: [Binary Subtraction Using 2's Complement 1 | Number System and Codes | Digital Circuit Design in EXTC](https://www.youtube.com/watch?v=K2oSFn7uRTY) — Ekeeda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/two-complement-111-110-nor-and
- Whole chapter: https://www.bctengineer.com/md/semester_2/basic_electronics_engineering/chapter-6

## Question

Subtract (111)2 from (110)2 using 2's complement method. Draw the circuit of AND gate using NOR gates only.

## Short answer

Subtraction: 2's complement of 111 is 001; 110 + 001 = 111, no carry, negative, magnitude 001 → −1. AND using NOR: two NOR gates as NOT invert A,B; a third NOR gates their outputs to give (A'+B')' = A·B.

## Answer

#### Subtraction: (110)₂ − (111)₂ using 2's complement

The question asks to subtract 111 from 110 (6 − 7), using 3 bits.

1. 1's complement of 111 = 000; 2's complement = 000 + 1 = **001**.
2. Add: 110 + 001 = **111**.
3. No end carry, so the result is negative.
4. Magnitude = 2's complement of 111 = 000 + 1 = 001.

**Answer: 110 − 111 = −(001)₂ = −1**

#### AND gate using only NOR gates

Using De Morgan's theorem: A·B = (A' + B')' = NOR(A', B').

- A NOR gate with both inputs joined acts as a NOT gate.
- So two NOR gates (as NOT) invert A and B, and a third NOR gate combines them.

```text
 A ---[NOR]--- A' ---+
      (inputs tied)  [NOR]---- Y = (A' + B')' = AB
 B ---[NOR]--- B' ---+
      (inputs tied)
```

Three 2-input NOR gates are needed.

## Check yourself

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

1. The 2's complement of 111 is:
   A. 001
   B. 000
   C. 110
   D. 010
   Answer: A. The answer states this: 001. (Key point: 2's complement)

2. 110 + 001 gives 111 with no end carry, so 110 - 111 equals:
   A. +1
   B. -1
   C. -7
   D. -6
   Answer: B. The answer states this: -1. (Key point: Result)

3. Using De Morgan, A.B equals:
   A. (A + B)'
   B. A' + B'
   C. (A' + B')'
   D. (A.B)'
   Answer: C. The answer states this: (A' + B')'. (Key point: AND from NOR)

4. An AND gate built only from NOR gates needs:
   A. Two 2-input NOR gates
   B. One 2-input NOR gate
   C. Four 2-input NOR gates
   D. Three 2-input NOR gates
   Answer: D. The answer states this: Three 2-input NOR gates. (Key point: NOR gate count)

## Related in Basic Electronics Engineering

- [X-OR gate using NAND gates only](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Axor-using-nand) — Basic Electronics Engineering, chapter 6
- [Why NOR and NAND are universal gates](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Auniversal-gates-nor-nand) — Basic Electronics Engineering, chapter 6
- [2's complement: 42 − 115](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Atwo-complement-42-115) — Basic Electronics Engineering, chapter 6
- [2's complement: (1111)2 − (1110)2](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Atwo-complement-1111-1110) — Basic Electronics Engineering, chapter 6
- [2's complement: (1111)2 − (1100)2](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Atwo-complement-1111-1100) — Basic Electronics Engineering, chapter 6

## Related in other subjects

- [3-input XNOR using NOR gates only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Athree-input-xnor-nor) — Digital Logic, chapter 2
- [2-input XOR using NOR gates only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Axor-using-nor-only) — Digital Logic, chapter 2, asked 2×
- [F = (A+B)(C+D)E using NOR gates only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Anor-only-function-realize) — Digital Logic, chapter 2
- [De Morgan; NOR gate from NAND gates](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-importance-nor-from-nand) — Digital Logic, chapter 2