---
title: "Commutative and associative laws; A+A'B = A+B"
subject: "Digital Logic"
chapter: 2
key: "semester_3/digital_logic:boolean-laws-simplify"
source: https://www.bctengineer.com/q/semester_3/digital_logic/boolean-laws-simplify
site: BCT Engineering
---

# Commutative and associative laws; A+A'B = A+B

Digital Logic · Chapter 2: Digital Logic

- Asked in: 2072 Chaitra, 2+2 marks
- Video: [Boolean Laws | Commutative, Associative, Distributive, AND, OR, Inversion Law | Digital Electronics](https://www.youtube.com/watch?v=6ehX4ZvWrDs) — Easy Engineering Classes
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/digital_logic/boolean-laws-simplify
- Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-2

## Question

Describe commutative and associative laws of Boolean algebra with examples and simplify A+A'B = A+B.

## Short answer

Commutative: A+B = B+A, AB = BA. Associative: A+(B+C) = (A+B)+C, A(BC) = (AB)C. Using distributive law, A + A'B = (A+A')(A+B) = 1·(A+B) = A+B.

## Answer

#### Commutative laws
The order of variables in an OR or AND operation does not change the result.
- $A + B = B + A$
- $A \cdot B = B \cdot A$

Example: $\bar A + C = C + \bar A$; with A = 1, B = 0: $1 \cdot 0 = 0 \cdot 1 = 0$. In hardware, swapping the inputs of an AND or OR gate does not change its output.

#### Associative laws
The grouping of variables in an OR or AND operation does not change the result.
- $A + (B + C) = (A + B) + C$
- $A \cdot (B \cdot C) = (A \cdot B) \cdot C$

Example: with A = 1, B = 0, C = 1: $1 \cdot (0 \cdot 1) = 0$ and $(1 \cdot 0) \cdot 1 = 0$. So a 3-input AND can be built from two 2-input AND gates in any order.

#### Proof of $A + \bar A B = A + B$
Using the distributive law $X + YZ = (X + Y)(X + Z)$:

$$
\begin{aligned} A + \bar A B &= (A + \bar A)(A + B) \\ &= 1 \cdot (A + B) \\ &= A + B \end{aligned}
$$

Truth-table check:

| A | B | $\bar A B$ | $A + \bar A B$ | $A + B$ |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 |

## Check yourself

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

1. Which expression shows the associative law of AND?
   A. A.(B.C) = (A.B).C
   B. A.B = B.A
   C. A.(B+C) = A.B + A.C
   D. A.A = A
   Answer: A. Associative law concerns grouping of variables, not their order. (Key point: Associative law)

2. What does the commutative law say about gate inputs?
   A. Grouping of three inputs can be changed
   B. Swapping inputs of an AND/OR gate does not change the output
   C. AND distributes over OR
   D. Output depends on input order in OR gates
   Answer: B. Commutativity: A+B = B+A and A.B = B.A. (Key point: Commutative law)

3. How is A + A'B = A + B proved algebraically?
   A. By De Morgan's law on A'B
   B. By the commutative law only
   C. Distributive law gives (A+A')(A+B) = 1.(A+B)
   D. By setting A = B
   Answer: C. Using X+YZ=(X+Y)(X+Z), A+A'B=(A+A')(A+B), and A+A'=1. (Key point: Proof step)

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