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

# Prove positive XOR equals negative XNOR

Digital Logic · Chapter 2: Digital Logic

- Asked in: 2070 Chaitra, 5 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/positive-xor-negative-xnor-prove
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-2

## Question

Prove that positive X-OR is equivalent to negative X-NOR.

## Short answer

In positive logic (H=1) the gate's output is XOR. In negative logic (H=0), H and L swap so the same physical gate yields 1 when inputs are equal, i.e., XNOR. Algebraically, Y_neg = (A' ⊕ B')' = A ⊙ B.

## Answer

In **positive logic** HIGH = 1 and LOW = 0; in **negative logic** HIGH = 0 and LOW = 1. Changing the convention complements every input and output bit of the same physical gate.

Take a gate that acts as XOR in positive logic. Its voltage table:

| 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 |

Y = 1 when inputs differ: **XOR**, $Y = A \oplus B$.

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

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

Rearranged in normal order (00, 01, 10, 11) the output is 1, 0, 0, 1, so Y = 1 when the inputs are equal: **XNOR**.

**Algebraic proof:** complement inputs and output of the positive-logic XOR:

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

Hence positive-logic XOR is equivalent to negative-logic XNOR.

## Check yourself

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

1. What changes when moving from positive to negative logic for the same gate?
   A. Only output bits are complemented
   B. Only inputs are complemented
   C. The supply voltage is reversed
   D. Every input and output bit is complemented
   Answer: D. HIGH=0 and LOW=1 in negative logic, so all bits are complemented. (Key point: Logic conventions)

2. A gate is XOR in positive logic. In negative logic its output is 1 when:
   A. The inputs are equal
   B. The inputs differ
   C. Both inputs are 1 only
   D. Both inputs are 0 only
   Answer: A. Rewriting in order gives 1,0,0,1, which is XNOR (1 when inputs are equal). (Key point: Negative-logic reading)

3. In the algebraic proof, Y_neg is obtained by:
   A. Complementing only the output of XOR
   B. Complementing both inputs and the output of the XOR
   C. Complementing only the inputs of XOR
   D. Adding a NOT after XNOR
   Answer: B. Y_neg = (A' xor B')' which simplifies to AB + A'B' = A xnor B. (Key point: Algebraic proof)

4. Hence positive-logic XOR equals negative-logic:
   A. XOR
   B. NAND
   C. XNOR
   D. AND
   Answer: C. The same physical gate behaves as XNOR in negative logic. (Key point: Final result)

## Close variants set in other exams

- [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 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
- [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's laws; XNOR using NOR gates](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-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×