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

# State and prove De Morgan's theorems

Basic Electronics Engineering · Chapter 6: Digital Electronics

- Asked in: 2073 Magh, 4 marks; 2069 Bhadra, 4 marks; 2067 Chaitra, 3 marks; 2074 Bhadra, 3 marks (asked 4 times)
- Video: [De Morgan's Theorem Explained: Basics, Statement, Circuit, and Proof](https://www.youtube.com/watch?v=OjQAwJx_DM0) — Engineering Funda
- 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-state-prove
- Whole chapter: https://www.bctengineer.com/md/semester_2/basic_electronics_engineering/chapter-6

## Question

State and prove De Morgan's theorems.

## Short answer

First theorem: (A + B)' = A'·B'. Second theorem: (A·B)' = A' + B'. Proof: a truth table shows columns for (A+B)' and A'·B' are identical, and columns for (A·B)' and A'+B' are identical, confirming the complement rules.

## Answer

#### Statements

1. **First theorem:** the complement of a sum (OR) equals the product (AND) of the complements.
   (A + B)' = A' · B'
2. **Second theorem:** the complement of a product equals the sum of the complements.
   (A · B)' = A' + B'

They extend to any number of variables: (A + B + C)' = A'B'C', (ABC)' = A' + B' + C'.

#### Proof by truth table

| A | B | A+B | (A+B)' | A' | B' | A'·B' | A·B | (A·B)' | A'+B' |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

The columns (A+B)' and A'·B' are identical, which proves the first theorem. The columns (A·B)' and A'+B' are identical, which proves the second theorem.

#### Algebraic check of the first theorem

Let X = A + B. Then X + X' = 1 and X·X' = 0. Take Y = A'B'.
- X + Y = A + B + A'B' = (A + A'B') + B = (A + B') + B = A + 1 = 1
- X·Y = (A + B)A'B' = AA'B' + BA'B' = 0 + 0 = 0

So Y is the complement of X, giving (A + B)' = A'B'.

## Check yourself

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

1. De Morgan's first theorem states:
   A. (A + B)' = A'.B'
   B. (A + B)' = A' + B'
   C. (A.B)' = A'.B'
   D. (A + B)' = A.B
   Answer: A. The answer states this: (A + B)' = A'.B'. (Key point: First theorem)

2. De Morgan's second theorem states:
   A. (A.B)' = A'.B'
   B. (A.B)' = A' + B'
   C. (A.B)' = A + B
   D. (A.B)' = (A + B)'
   Answer: B. The answer states this: (A.B)' = A' + B'. (Key point: Second theorem)

3. How does the truth table prove the theorem?
   A. The columns are complements of each other
   B. Only the first row matches
   C. The columns of the two sides are identical for all inputs
   D. Only the last row matches
   Answer: C. The answer states this: The columns of the two sides are identical for all inputs. (Key point: Truth table proof)

4. In the algebraic proof, Y = A'B' is shown to be the complement of X = A + B because:
   A. X + Y = 0 and X.Y = 1
   B. X + Y = X.Y
   C. X = Y
   D. X + Y = 1 and X.Y = 0
   Answer: D. The answer states this: X + Y = 1 and X.Y = 0. (Key point: Algebraic proof method)

## Related in Basic Electronics Engineering

- [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

## Related in other subjects

- [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×
- [State and prove De Morgan's laws](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-state-prove) — Digital Logic, chapter 2, asked 2×
- [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