---
title: "De Morgan's theorem and duality theorem"
subject: "Basic Electronics Engineering"
chapter: 6
key: "semester_2/basic_electronics_engineering:de-morgan-duality"
source: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/de-morgan-duality
site: BCT Engineering
---

# De Morgan's theorem and duality theorem

Basic Electronics Engineering · Chapter 6: Digital Electronics

- Asked in: 2072 Magh, 4 marks
- Video: [DeMorgan's Theorem and Principle of Duality with - Proof | English](https://www.youtube.com/watch?v=HAqaZab7tr8) — Dopamine
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/de-morgan-duality
- Whole chapter: https://www.bctengineer.com/md/semester_2/basic_electronics_engineering/chapter-6

## Question

State De Morgan's theorem and Duality theorem with two examples for each.

## Short answer

De Morgan's theorems: (A+B)' = A'·B' and (A·B)' = A'+B'. Duality theorem: interchange AND/OR and 0/1; dual of A+0=A is A·1=A, dual of A+AB=A is A(A+B)=A.

## Answer

#### De Morgan's theorems

1. The complement of a sum equals the product of the complements: **(A + B)' = A'·B'**
2. The complement of a product equals the sum of the complements: **(A·B)' = A' + B'**

Example 1: (X + Y + Z)' = X'Y'Z'. For X = 1, Y = 0, Z = 0: LHS = (1)' = 0, RHS = 0·1·1 = 0.

Example 2: (A'B)' = A + B'. Check: (A·B)' with A replaced by A' gives (A')' + B' = A + B'.

#### Duality theorem

Every Boolean identity remains valid if we interchange AND with OR, and 0 with 1 (variables are not changed). The new identity is the **dual** of the original.

Example 1:
- A + 0 = A has dual A · 1 = A.
- A + A' = 1 has dual A · A' = 0.

Example 2:
- A + AB = A has dual A(A + B) = A.
- A + A'B = A + B has dual A(A' + B) = AB.

Duality is useful because proving one law automatically proves its dual.

## Check yourself

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

1. The complement of a sum equals:
   A. The product of the complements
   B. The sum of the complements
   C. The complement of the product
   D. The product of the variables
   Answer: A. The answer states this: The product of the complements. (Key point: De Morgan statement)

2. (A'B)' equals:
   A. A' + B
   B. A + B'
   C. A'B'
   D. A + B
   Answer: B. The answer states this: A + B'. (Key point: De Morgan example)

3. To form the dual of a Boolean identity, we:
   A. Complement every variable
   B. Interchange variables A and B
   C. Interchange AND with OR and 0 with 1, leaving variables unchanged
   D. Replace every 1 with 0 only
   Answer: C. The answer states this: Interchange AND with OR and 0 with 1, leaving variables unchanged. (Key point: Duality rule)

4. The dual of A + A' = 1 is:
   A. A . A' = 1
   B. A + A' = 0
   C. A' . A = 1
   D. A . A' = 0
   Answer: D. The answer states this: A . A' = 0. (Key point: Dual of A + A' = 1)

## Related in Basic Electronics Engineering

- [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×

## Related in other subjects

- [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/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
- [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
- [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