Postulates and Theorems of Boolean Algebra : Postulates and theorems form the rules for simplifying Boolean expressions. Key laws include identity, complement, associative, distributive, and De Morgan’s Theorems. They help design and reduce logic circuits efficiently.
Postulates of Boolean Algebra
Boolean Algebra is built on a set of rules called postulates. These are like the foundation rules we use to handle binary values (0 and 1). Let’s break them down….
1. Identity Law
- A + 0 = A
- A · 1 = A
Explanation: Adding 0 to anything doesn’t change it. Multiplying anything by 1 keeps it the same.
Example:
If A = 1 →
A + 0 = 1 + 0 = 1
A · 1 = 1 · 1 = 1
2. Null Law
- A + 1 = 1
- A · 0 = 0
Explanation: Adding 1 to anything makes it 1. Multiplying by 0 always gives 0.
Example:
If A = 0 →
A + 1 = 0 + 1 = 1
A · 0 = 0 · 0 = 0
3. Complement Law
- A + A′ = 1
- A · A′ = 0
Explanation: A variable and its complement (opposite) always make either 1 (OR) or 0 (AND).
Example:
If A = 1, then A′ = 0 →
A + A′ = 1 + 0 = 1
A · A′ = 1 · 0 = 0
4. Idempotent Law
- A + A = A
- A · A = A
Explanation: Repeating a value with OR or AND does not change it.
Example:
A = 1 → A + A = 1 + 1 = 1; A · A = 1 · 1 = 1
5. Domination Law (Also called Annulment)
- A + 1 = 1
- A · 0 = 0
Explanation: OR with 1 will always result in 1. AND with 0 will always result in 0.
6. Double Negation
- (A′)′ = A
Explanation: Taking the complement twice gives the original value back.
Theorems of Boolean Algebra
Theorems are derived from postulates. These help simplify Boolean expressions easily. Basic Postulates and theorems of Boolean algebra
1. Commutative Law
- A + B = B + A
- A · B = B · A
Explanation: Order doesn’t matter in AND or OR operations.
2. Associative Law
- A + (B + C) = (A + B) + C
- A · (B · C) = (A · B) · C
Explanation: You can group variables however you want. The result remains the same.
3. Distributive Law
- A · (B + C) = A·B + A·C
- A + (B · C) = (A + B) · (A + C)
Explanation: Similar to multiplication over addition in normal algebra, it distributes.
Example:
Let A = 1, B = 0, C = 1
Then:
A · (B + C) = 1 · (0 + 1) = 1 · 1 = 1
A·B + A·C = 1·0 + 1·1 = 0 + 1 = 1
4. Absorption Law
- A + A·B = A
- A · (A + B) = A
Explanation: Extra terms can be removed without changing the logic.
Example:
A = 1, B = 0
A + A·B = 1 + 1·0 = 1 + 0 = 1
A · (A + B) = 1 · (1 + 0) = 1 · 1 = 1
5. De Morgan’s Theorems
These are very important in digital circuit design.
- (A · B)′ = A′ + B′
- (A + B)′ = A′ · B′
Explanation: The NOT operation flips the logic, and changes AND to OR or vice versa.
Example:
A = 1, B = 0
(A + B)’ = (1 + 0)’ = 1′ = 0
A’ · B’ = 0 · 1 = 0
Why Are Postulates and Theorems Laws Useful?
These postulates and theorems are used to:
- Simplify logic circuits
- Reduce the number of gates in a circuit
- Design efficient digital systems
- Analyze logical conditions and create truth tables
Example:
Expression:
A + A·B
Using Absorption Law:
= A
Truth Table:
| A | B | A·B | A + A·B |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
Compare this with A. You’ll see both columns are the same
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