---
title: "Positive XOR equals negative XNOR"
subject: "Digital Logic"
chapter: 2
key: "semester_3/digital_logic:positive-xor-negative-xnor-define"
source: https://www.bctengineer.com/q/semester_3/digital_logic/positive-xor-negative-xnor-define
site: BCT Engineering
---

# Positive XOR equals negative XNOR

Digital Logic · Chapter 2: Digital Logic

- Asked in: 2079 Bhadra, 2+3 marks; 2079 Baisakh, 6 marks (asked 2 times)
- 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/positive-xor-negative-xnor-define
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-2

## Question

Define positive and negative logic and prove that positive X-OR is equivalent to negative X-NOR.

## Short answer

Positive logic assigns high voltage as 1, low as 0; negative logic reverses this. The same voltage table that gives XOR in positive logic yields XNOR in negative logic because the function becomes complement of XOR with complemented inputs, resulting in A⊙B.

## Answer

#### Positive and negative logic
- **Positive logic:** the higher voltage level represents logic 1 and the lower level represents logic 0. Example in TTL: +5 V = 1, 0 V = 0.
- **Negative logic:** the higher voltage level represents logic 0 and the lower level represents logic 1. Example: +5 V = 0, 0 V = 1 (or in a –5 V system, –5 V = 1, 0 V = 0).

The same physical circuit gives a different logic function depending on which convention is used. Changing from positive to negative logic means complementing **every** input and output bit.

#### Proof: positive XOR = negative XNOR
Take a gate working as XOR in positive logic. Its voltage table (H = high, L = low):

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

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

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

This is XOR: $Y = A \oplus B$.

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

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

Output is 1 when both inputs are equal, which is XNOR: $Y = \overline{A \oplus B}$.

**Algebraic proof:** in negative logic the function becomes the complement of positive XOR with complemented inputs:

$$
\begin{aligned} Y_{neg} &= \overline{\bar A \oplus \bar B} \\ &= \overline{\bar A B + A \bar B} \\ &= \overline{A \oplus B} = A \odot B \end{aligned}
$$

(since $\bar A \oplus \bar B = A \oplus B$). Hence the **positive-logic XOR gate is the same circuit as a negative-logic XNOR gate**.

## Check yourself

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

1. What is positive logic?
   A. Higher voltage is 0
   B. Higher voltage is 1, lower is 0
   C. Always +5 V is 0
   D. Only negative voltages
   Answer: B. Negative logic is the reverse. (Key point: Positive logic)

2. What happens when changing from positive to negative logic?
   A. Every input and output bit is complemented
   B. Only the output is complemented
   C. Nothing changes
   D. Only inputs change
   Answer: A. The same circuit realises a dual function. (Key point: Effect of change)

3. Negative-logic output of a positive XOR gate is 1 when?
   A. Inputs differ
   B. Both are 1 only
   C. Both are 0 only
   D. Both inputs are equal
   Answer: D. This is XNOR. (Key point: Truth table result)

4. Why does the algebra work?
   A. XOR is the same as AND
   B. XOR is always 0
   C. Complementing both inputs of XOR leaves it unchanged (A'xorB' = AxorB)
   D. NOT does not affect XOR output
   Answer: C. So Yneg = (A xor B)'. (Key point: Identity)

## Close variants set in other exams

- [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×

## 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
- [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×
- [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
- [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; verify 3-variable De Morgan](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-three-variable-verify) — Digital Logic, chapter 2