Here is a clear, simple, and exam-oriented explanation of Boolean Algebra—perfect for BCA/MCA/engineering mathematics and discrete structures exams.
⭐ Boolean Algebra – Introduction
Boolean Algebra is an algebraic structure used to work with logical values (True/False or 1/0).
It forms the mathematical foundation of:
✔ Digital logic
✔ Computer circuits
✔ Switching theory
✔ Logic design
✔ Algorithms
✔ Set theory
Created by George Boole (1854).
⭐ Boolean Algebra Definition
A Boolean Algebra is a set B with two binary operations:
- + (OR)
- · (AND)
and a unary operation:
- ’ (NOT / complement)
together with two special elements:
- 0 (identity for OR)
- 1 (identity for AND)
This structure satisfies a set of axioms/laws.
Typical Boolean algebra uses elements {0,1}.
⭐ Boolean Variables
A Boolean variable takes either value:
- 0 → False
- 1 → True
Example:
A, B, C are Boolean variables ⇒ A ∈ {0,1}
⭐ Basic Boolean Operations
✔ 1. OR Operation (+)
| A | B | A + B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Equivalent to logical OR, set union, electrical parallel.
✔ 2. AND Operation (·)
| A | B | A · B |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Equivalent to logical AND, set intersection, electrical series.
✔ 3. NOT Operation ( A’ )
| A | A’ |
|---|---|
| 0 | 1 |
| 1 | 0 |
Equivalent to logical NOT, set complement.
⭐ Important Boolean Algebra Laws
Boolean algebra follows 10 main types of laws.
1️⃣ Commutative Laws
- A + B = B + A
- A · B = B · A
2️⃣ Associative Laws
- (A + B) + C = A + (B + C)
- (A · B) · C = A · (B · C)
3️⃣ Distributive Laws
(Unlike regular algebra!)
- A · (B + C) = AB + AC
- A + BC = (A + B)(A + C)
4️⃣ Identity Laws
- A + 0 = A
- A · 1 = A
5️⃣ Null Laws
- A + 1 = 1
- A · 0 = 0
6️⃣ Idempotent Laws
- A + A = A
- A · A = A
7️⃣ Complement Laws
- A + A’ = 1
- A · A’ = 0
8️⃣ Involution Law
- (A’)’ = A
9️⃣ Absorption Laws
- A + AB = A
- A(A + B) = A
🔟 De Morgan’s Laws
- (A + B)’ = A’B’
- (A · B)’ = A’ + B’
These are essential for simplification.
⭐ Duality Principle
The duality principle states:
If you interchange + ↔ · and 0 ↔ 1 in any Boolean identity, the resulting identity is also valid.
Example:
A + 0 = A
Dual: A · 1 = A
⭐ Boolean Algebra and Set Theory
Boolean algebra corresponds to set operations:
| Boolean | Sets | Logical |
|---|---|---|
| A + B | A ∪ B | A OR B |
| AB | A ∩ B | A AND B |
| A’ | Aᶜ | NOT A |
⭐ Boolean Expressions
A Boolean expression combines variables and operators.
Example:
F = AB + A’C
Goal in digital logic: simplify expressions to reduce circuit size.
⭐ Canonical Forms
1. Sum of Products (SOP)
Expression in OR of AND terms.
Example:
F = A’B + AB’
2. Product of Sums (POS)
Expression in AND of OR terms.
Example:
F = (A + B’)(A’ + C)
⭐ Boolean Functions
A Boolean function maps:
[
f : {0,1}^n \to {0,1}
]
Examples:
- f(A) = A’
- f(A, B) = AB + A’B’
⭐ Applications
- Logic gates
- Digital circuits
- Simplification using Karnaugh maps (K-maps)
- Computer design
- Automata
- Switching theory
- Database queries
- Software logical conditions
⭐ Quick Exam Summary
Boolean Algebra:
A system with elements {0,1}, operations (+,·,’), satisfying a set of laws.
Main operations:
AND, OR, NOT.
Main laws:
Commutative
Associative
Distributive
Identity
Null
Idempotent
Complement
Absorption
De Morgan’s
Forms:
SOP, POS
Canonical forms
Applications:
Digital logic and circuit design.