---
title: "Positive/negative logic; verify 3-variable De Morgan"
subject: "Digital Logic"
chapter: 2
key: "semester_3/digital_logic:demorgan-three-variable-verify"
source: https://www.bctengineer.com/q/semester_3/digital_logic/demorgan-three-variable-verify
site: BCT Engineering
---

# Positive/negative logic; verify 3-variable De Morgan

Digital Logic · Chapter 2: Digital Logic

- Asked in: 2082 Baisakh, 2+4 marks
- Video: [Positive and Negative Logic](https://www.youtube.com/watch?v=gtucOKD1pHA) — Neso Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/demorgan-three-variable-verify
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-2

## Question

Define the positive and negative logic. State and verify 3-bits De Morgan's theorem.

## Short answer

Positive logic assigns HIGH=1, LOW=0; negative logic reverses. De Morgan’s law for three variables: (A+B+C)′ = A′B′C′ and (ABC)′ = A′+B′+C′. Truth table verification shows the corresponding columns are identical for all eight input combinations, confirming both theorems.

## Answer

#### Positive and negative logic
- **Positive logic:** HIGH voltage = logic 1, LOW voltage = logic 0 (e.g. TTL: +5 V = 1, 0 V = 0).
- **Negative logic:** HIGH voltage = logic 0, LOW voltage = logic 1 (e.g. +5 V = 0, 0 V = 1).

The same physical gate performs a different logic function in the two systems, because changing the convention complements every input and output. Example: a gate whose output is HIGH only when both inputs are HIGH is an AND gate in positive logic but an OR gate in negative logic.

#### De Morgan's theorem for 3 variables
**Theorem 1:** $\overline{A + B + C} = \bar A \cdot \bar B \cdot \bar C$

**Theorem 2:** $\overline{A \cdot B \cdot C} = \bar A + \bar B + \bar C$

**Verification by truth table**

| A | B | C | $\overline{A+B+C}$ | $\bar A \bar B \bar C$ | $\overline{ABC}$ | $\bar A + \bar B + \bar C$ |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |

Columns 4 and 5 are identical, and columns 6 and 7 are identical for all 8 combinations, so both theorems hold for 3 variables.

**Algebraic check** (using the 2-variable law with $X = B + C$):

$$
\begin{aligned} \overline{A + B + C} &= \overline{A + X} = \bar A \cdot \bar X \\ &= \bar A \cdot \overline{B + C} = \bar A \bar B \bar C \end{aligned}
$$

Similarly, $\overline{A(BC)} = \bar A + \overline{BC} = \bar A + \bar B + \bar C$.

```text
 3-input NOR = bubbled 3-input AND
 3-input NAND = bubbled 3-input OR
```

## Check yourself

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

1. Negative logic defines:
   A. HIGH = 1, LOW = 0
   B. HIGH = LOW
   C. HIGH = 0, LOW = 1
   D. Only negative voltage
   Answer: C. This is the opposite of positive logic. (Key point: Negative logic)

2. A gate that is AND in positive logic is what in negative logic?
   A. NAND
   B. XOR
   C. NOT
   D. OR
   Answer: D. Output HIGH only when both inputs HIGH becomes OR for the complemented convention. (Key point: AND/OR switch)

3. 3-variable De Morgan's first theorem:
   A. (A+B+C)' = A'.B'.C'
   B. (A+B+C)' = A'+B'+C'
   C. (ABC)' = A'B'C'
   D. (A+B+C)' = ABC
   Answer: A. Complement of a sum equals product of complements. (Key point: Theorem 1)

4. The algebraic check treats B+C as:
   A. A constant 1
   B. A single variable X to reuse the 2-variable law
   C. A complement
   D. A NAND
   Answer: B. With X=B+C, (A+X)' = A'X' = A'B'C'. (Key point: Algebraic proof)

## Related in Digital Logic

- [De Morgan; positive NAND equals negative NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-positive-nand-negative-nor) — Digital Logic, chapter 2
- [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
- [De Morgan's theorem and (-47+27) by 2's complement](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-twos-complement-addition) — Digital Logic, chapter 2
- [Positive and negative logic](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Apositive-negative-logic) — Digital Logic, chapter 2, asked 2×
- [Positive/negative logic; XNOR using NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Apositive-negative-logic-xnor-nand) — Digital Logic, chapter 2, asked 2×
- [Positive/negative logic; Ex-NOR using NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Apositive-negative-logic-xnor-nor) — Digital Logic, chapter 2, asked 2×

## Related in other subjects

- [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
- [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×
- [Effective mass can be positive or negative](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aeffective-mass-positive-negative) — Electrical Engineering Material, chapter 1, asked 4×
- [Short note: Positive and negative feedback](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Ashort-note-feedback) — Basic Electronics Engineering, chapter 4