Table of Contents
Introduction
Boolean Algebra is a branch of mathematics used in digital electronics and computer science. It deals with only two values:
👉 0 (False)
👉 1 (True)
These laws help simplify logical expressions and design circuits.
Basic Boolean Laws
1. Identity Law
- A + 0 = A
- A · 1 = A
👉 Meaning: Adding 0 or multiplying by 1 does not change the value.
Example:
If A = 1
→ 1 + 0 = 1
→ 1 · 1 = 1
2. Null Law (Domination Law)
- A + 1 = 1
- A · 0 = 0
👉 Meaning:
- OR with 1 always gives 1
- AND with 0 always gives 0
3. Idempotent Law
- A + A = A
- A · A = A
👉 Meaning: Repeating the same variable does not change the result.
4. Complement Law
- A + A′ = 1
- A · A′ = 0
👉 Meaning: A variable and its complement always give fixed results.
5. Double Negation Law
- (A′)′ = A
👉 Meaning: Double NOT returns the original value.
6. Commutative Law
- A + B = B + A
- A · B = B · A
👉 Meaning: Order does not matter.
7. Associative Law
- (A + B) + C = A + (B + C)
- (A · B) · C = A · (B · C)
👉 Meaning: Grouping does not matter.
8. Distributive Law
- A · (B + C) = (A · B) + (A · C)
- A + (B · C) = (A + B) · (A + C)
👉 Very important for simplification.
9. Absorption Law
- A + (A · B) = A
- A · (A + B) = A
👉 Meaning: Extra terms can be absorbed.
10. De Morgan’s Theorems ⭐ (Very Important)
- (A · B)′ = A′ + B′
- (A + B)′ = A′ · B′
👉 Used in logic circuit design.
Summary Table
| Law | Expression |
|---|---|
| Identity | A + 0 = A |
| Null | A + 1 = 1 |
| Idempotent | A + A = A |
| Complement | A + A′ = 1 |
| Double Negation | (A′)′ = A |
| Commutative | A + B = B + A |
| Associative | (A + B) + C = A + (B + C) |
| Distributive | A(B + C) = AB + AC |
| Absorption | A + AB = A |
| De Morgan | (AB)’ = A’ + B’ |
Example for Practice
Simplify:
A + (A · B)
Using Absorption Law:
👉 Answer = A
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