---
title: "State and prove De Morgan's laws"
subject: "Digital Logic"
chapter: 2
key: "semester_3/digital_logic:demorgan-state-prove"
source: https://www.bctengineer.com/q/semester_3/digital_logic/demorgan-state-prove
site: BCT Engineering
---

# State and prove De Morgan's laws

Digital Logic · Chapter 2: Digital Logic

- Asked in: 2082 Shrawan, 2 marks; 2076 Chaitra, 2 marks (asked 2 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_3/digital_logic/demorgan-state-prove
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-2

## Question

State and prove De-Morgan's laws.

## Short answer

De Morgan's laws: ¬(A+B) = ¬A·¬B and ¬(A·B) = ¬A+¬B. Proof by truth table shows both sides match for all four input combinations, confirming the identities.

## Answer

**First theorem:** the complement of a sum (OR) of variables equals the product (AND) of their complements.
$$
\overline{A + B} = \bar{A} \cdot \bar{B}
$$

**Second theorem:** the complement of a product (AND) of variables equals the sum (OR) of their complements.
$$
\overline{A \cdot B} = \bar{A} + \bar{B}
$$

In words: "break the bar and change the sign".

**Proof by truth table**

| A | B | $\overline{A+B}$ | $\bar A \cdot \bar B$ | $\overline{AB}$ | $\bar A + \bar B$ |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |

For all four input combinations, column 3 equals column 4 and column 5 equals column 6, so both theorems are proved.

**Algebraic proof (Theorem 1):** let $X = A + B$ and $Y = \bar A \bar B$. Then $X + Y = A + B + \bar A \bar B = A + B + \bar B = 1$ and $X Y = A\bar A \bar B + B \bar A \bar B = 0$. A variable that ORs with X to give 1 and ANDs with X to give 0 is $\bar X$, so $\overline{A+B} = \bar A \bar B$. Theorem 2 follows by duality.

## Check yourself

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

1. (A+B)' =
   A. A'B'
   B. A'+B'
   C. AB
   D. A+B'
   Answer: A. Break the bar and change the sign. (Key point: Theorem 1)

2. (AB)' =
   A. A'+B'
   B. A'B'
   C. A+B
   D. AB'
   Answer: A. Complement of a product is sum of complements. (Key point: Theorem 2)

3. How is truth table proof concluded?
   A. One row matches
   B. Output is always 1
   C. Inputs are removed
   D. Columns match for all input combinations
   Answer: D. All four rows must agree. (Key point: Truth table proof)

4. What shows Y is the complement of X?
   A. X+Y = 0 and XY = 1
   B. X = Y
   C. X+Y = 1 and XY = 0
   D. X+Y = X
   Answer: C. That uniquely defines the complement. (Key point: Algebraic proof)

## Related in Digital Logic

- [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
- [De Morgan's laws; XOR using 3-input NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-xor-three-input-nand) — Digital Logic, chapter 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
- [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×
- [De Morgan; NOR gate from NAND gates](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-importance-nor-from-nand) — Digital Logic, chapter 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

## Related in other subjects

- [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×
- [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