Boolean Algebra Simplifier
Simplify Boolean expressions using algebraic laws and theorems.
Basic Operations
Boolean algebra uses three basic operations:
- AND (⋅): The output is true only when all inputs are true.
- OR (+): The output is true when at least one input is true.
- NOT (¯): The output is the inverse of the input.
Basic Laws
| Law | AND Form | OR Form |
|---|---|---|
| Identity | A ⋅ 1 = A | A + 0 = A |
| Null | A ⋅ 0 = 0 | A + 1 = 1 |
| Idempotent | A ⋅ A = A | A + A = A |
| Complement | A ⋅ Ā = 0 | A + Ā = 1 |
| Commutative | A ⋅ B = B ⋅ A | A + B = B + A |
| Associative | (A ⋅ B) ⋅ C = A ⋅ (B ⋅ C) | (A + B) + C = A + (B + C) |
| Distributive | A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C) | A + (B ⋅ C) = (A + B) ⋅ (A + C) |
| Absorption | A ⋅ (A + B) = A | A + (A ⋅ B) = A |
| De Morgan's | (A ⋅ B)̅ = Ā + B̅ | (A + B)̅ = Ā ⋅ B̅ |
Simplification Example
Problem: Simplify (A + B) ⋅ (A + C)
Step 1: Apply the distributive law.
(A + B) ⋅ (A + C) = A ⋅ A + A ⋅ C + B ⋅ A + B ⋅ C
Step 2: Apply the idempotent law (A ⋅ A = A).
= A + A ⋅ C + B ⋅ A + B ⋅ C
Step 3: Apply the absorption law (A + A ⋅ C = A).
= A + B ⋅ A + B ⋅ C
Step 4: Apply the commutative law (B ⋅ A = A ⋅ B).
= A + A ⋅ B + B ⋅ C
Step 5: Apply the absorption law again.
= A + B ⋅ C
Final Answer: A + (B ⋅ C)
Boolean Algebra in Computer Science
Foundation of Digital Systems
Boolean algebra forms the mathematical foundation of digital systems and computer architecture. It's used to design and analyze digital circuits, from simple logic gates to complex microprocessors.
Circuit Optimization
Simplifying Boolean expressions is crucial for optimizing digital circuits. Simpler expressions lead to circuits with fewer gates, which are faster, consume less power, and are cheaper to manufacture.
A-Level Exam Tips
In A-Level Computer Science exams, you may be asked to simplify Boolean expressions, convert between Boolean expressions and logic circuits, or apply Boolean algebra to solve problems. Remember the key laws and practice applying them!