---
title: "De Morgan; positive NAND equals negative NOR"
subject: "Digital Logic"
chapter: 2
key: "semester_3/digital_logic:demorgan-positive-nand-negative-nor"
source: https://www.bctengineer.com/q/semester_3/digital_logic/demorgan-positive-nand-negative-nor
site: BCT Engineering
---

# De Morgan; positive NAND equals negative NOR

Digital Logic · Chapter 2: Digital Logic

- Asked in: 2079 Bhadra, 2+4 marks
- Video: [positive logic nand gate is equivalent to negative logic nor gate](https://www.youtube.com/watch?v=-Cd5Zt5XaoM) — Aasaan padhaai
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/demorgan-positive-nand-negative-nor
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-2

## Question

State and prove De-Morgan's theorems with necessary diagrams and prove that positive NAND equivalent is equal to negative NOR.

## Short answer

De Morgan's laws: ¬(A+B)=¬A·¬B, ¬(A·B)=¬A+¬B, proved by truth table. A positive NAND (output 0 only when both inputs 1) becomes a NOR under negative logic because complementing all signals gives output 1 only when both inputs 0, matching the NOR function.

## Answer

#### De Morgan's theorems
**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".

**Proof by truth table**

| A | B | $\overline{A+B}$ | $\bar A \cdot \bar B$ | $\overline{AB}$ | $\bar A + \bar B$ |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |

For all four input combinations, column 3 equals column 4 and column 5 equals column 6, so both theorems are proved.

**Diagrams**

```text
 Theorem 1: NOR = bubbled AND
 A --|‾‾‾\             A --o|‾‾‾\
     | OR  )o-- Y   =       | AND )-- Y
 B --|___/             B --o|___/

 Theorem 2: NAND = bubbled OR
 A --|‾‾‾\             A --o|‾‾‾\
     | AND )o-- Y   =       | OR  )-- Y
 B --|___/             B --o|___/

 (o = inversion bubble)
```

#### Positive NAND = negative NOR
Consider a gate whose voltage table is:

| A | B | Y |
|---|---|---|
| L | L | H |
| L | H | H |
| H | L | H |
| H | H | L |

**Positive logic** (H = 1, L = 0):

| A | B | Y |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |

Output is 0 only when both inputs are 1: a **NAND** gate.

**Negative logic** (H = 0, L = 1), same circuit:

| A | B | Y |
|---|---|---|
| 1 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |

Output is 1 only when both inputs are 0: a **NOR** gate.

**Algebraically:** going to negative logic complements all inputs and the output:

$$
\begin{aligned} Y_{neg} &= \overline{\,\overline{\bar A \cdot \bar B}\,} \\ &= \bar A \cdot \bar B \\ &= \overline{A + B} \quad (\text{De Morgan}) \end{aligned}
$$

Hence positive-logic NAND is equivalent to negative-logic NOR.

## Check yourself

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

1. NOR gate equals?
   A. Bubbled OR
   B. Plain AND
   C. Plain NAND
   D. Bubbled AND
   Answer: D. (A+B)' = A'B'. (Key point: NOR equivalence)

2. Output L only when?
   A. Both L
   B. Any input H
   C. Inputs differ
   D. Both inputs H
   Answer: D. That is NAND in positive logic. (Key point: NAND voltage table)

3. Same circuit in negative logic?
   A. NAND
   B. AND
   C. NOR
   D. XOR
   Answer: C. Output 1 only when both inputs 0. (Key point: Negative logic)

4. Yneg =
   A. AB
   B. A'B' = (A+B)'
   C. A+B
   D. (AB)'
   Answer: B. Complement all inputs and output, then use De Morgan. (Key point: Algebra)

## Related in Digital Logic

- [De Morgan; negative OR equals positive AND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-negative-or-positive-and) — Digital Logic, chapter 2
- [Prove De Morgan's theorems; XOR using NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-prove-xor-nand) — Digital Logic, chapter 2, asked 3×
- [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
- [Prove positive XOR equals negative XNOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Apositive-xor-negative-xnor-prove) — Digital Logic, chapter 2, asked 3×
- [Positive/negative logic; verify 3-variable De Morgan](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-three-variable-verify) — Digital Logic, chapter 2
- [Positive XOR equals negative XNOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Apositive-xor-negative-xnor-define) — Digital Logic, chapter 2, asked 3×

## Related in other subjects

- [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×
- [Equal positive and negative sequence impedances](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Astatic-circuit-positive-negative-equal) — Power System Analysis II, chapter 4