---
title: "De Morgan's laws; XNOR using NOR gates"
subject: "Digital Logic"
chapter: 2
key: "semester_3/digital_logic:demorgan-xnor-nor"
source: https://www.bctengineer.com/q/semester_3/digital_logic/demorgan-xnor-nor
site: BCT Engineering
---

# De Morgan's laws; XNOR using NOR gates

Digital Logic · Chapter 2: Digital Logic

- Asked in: 2081 Bhadra, 2+3 marks
- Video: [Implementation of all gates using NOR gate | Design with Universal gates | Digital system design](https://www.youtube.com/watch?v=rfaKzj5go-0) — RF Design Basics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/demorgan-xnor-nor
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-2

## Question

State De Morgan's Laws. Construct 2 input XNOR gate using only NOR gates.

## Short answer

De Morgan’s laws: (A+B)′ = A′B′ and (AB)′ = A′+B′. A 2-input XNOR gate can be realized with four NOR gates: G1 = (A+B)′, G2 = (A+G1)′, G3 = (B+G1)′, then Y = (G2+G3)′ = (A′B+AB′)′ = A⊙B, matching the equivalence function.

## Answer

#### De Morgan's laws
**First theorem:** the complement of a sum (OR) of variables equals the product (AND) of their complements.
$$
\overline{A + B} = \bar{A} \cdot \bar{B}
$$

**Second theorem:** the complement of a product (AND) of variables equals the sum (OR) of their complements.
$$
\overline{A \cdot B} = \bar{A} + \bar{B}
$$

In words: "break the bar and change the sign".

Example: $\overline{\bar A + B} = A \bar B$ and $\overline{A \bar C} = \bar A + C$.

#### 2-input XNOR using only NOR gates
$Y = A \odot B = AB + \bar A \bar B = \overline{A\bar B + \bar A B}$ needs **four 2-input NOR gates** (the dual of the 4-NAND XOR).

$$
\begin{aligned} G_1 &= \overline{A + B} \\ G_2 &= \overline{A + G_1} = \bar A (A + B) = \bar A B \\ G_3 &= \overline{B + G_1} = \bar B (A + B) = A \bar B \\ Y &= \overline{G_2 + G_3} = \overline{\bar A B + A \bar B} = A \odot B \end{aligned}
$$

```text
 A --+---------------[NOR G2]--+
     |                 |       |
     +--[NOR G1]---G1--+       +--[NOR G4]-- Y
     |                 |       |
 B --+---------------[NOR G3]--+
```

| A | B | $G_1$ | $G_2$ | $G_3$ | Y |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 | 1 |

Y = 1 only when A = B, which is the XNOR function.

## Check yourself

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

1. De Morgan's second theorem:
   A. (AB)' = A'B'
   B. (A+B)' = A'+B'
   C. (AB)' = A' + B'
   D. (AB)' = A+B
   Answer: C. Complement of a product equals the sum of complements. (Key point: Statement)

2. How many 2-input NOR gates make an XNOR?
   A. 3
   B. 5
   C. 2
   D. 4
   Answer: D. G1, G2, G3 and the output NOR. (Key point: XNOR gates)

3. G2 = NOR(A, G1) simplifies to:
   A. A'B
   B. AB'
   C. A'B'
   D. AB
   Answer: A. G2 = A'(A+B) = A'B. (Key point: G2 value)

4. The output Y is 1 when:
   A. A differs from B
   B. A = B
   C. Both are 0 only
   D. A = 1 only
   Answer: B. Y = (A'B + AB')' = XNOR. (Key point: Output)

## Related in Digital Logic

- [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
- [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
- [De Morgan's laws; XOR using 3-input NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-xor-three-input-nand) — Digital Logic, chapter 2
- [De Morgan's laws; XOR from NAND gates](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-importance-xor-nand) — Digital Logic, chapter 2
- [De Morgan; X-NOR using a universal gate](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-xnor-universal-gate) — 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×

## Related in other subjects

- [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
- [State and prove De Morgan's theorems](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Ade-morgan-state-prove) — Basic Electronics Engineering, chapter 6, asked 4×
- [De Morgan's theorem and duality theorem](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Ade-morgan-duality) — Basic Electronics Engineering, chapter 6