Overview
Boolean algebra is the mathematical framework for digital logic, named after mathematician George Boole who developed it in 1847. It operates on binary variables (values of 0 or 1) using three fundamental operations: AND (·), OR (+), and NOT (complement, '). Claude Shannon's 1937 thesis demonstrated that boolean algebra perfectly models switching circuits, founding the entire field of digital electronics.
Mastering boolean algebra rules lets you simplify complex logic expressions — reducing the number of gates in a circuit, decreasing manufacturing cost, lowering power consumption, and improving circuit speed. The rules are not arbitrary; they reflect the fundamental properties of binary variables and logical operations that have been rigorously proven.
This page is a complete reference for all boolean algebra laws and theorems, with worked examples showing how to apply them for circuit simplification.
How It Works
The foundational boolean algebra laws are:
BASIC LAWS: Identity: A + 0 = A | A · 1 = A Null: A + 1 = 1 | A · 0 = 0 Idempotent: A + A = A | A · A = A Complement: A + A' = 1 | A · A' = 0 Double Negation: (A')' = A
COMBINATION LAWS: 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) = AB + AC | A+(B·C) = (A+B)·(A+C)
DE MORGAN'S THEOREMS: (A · B)' = A' + B' (complement of AND = OR of complements) (A + B)' = A' · B' (complement of OR = AND of complements)
ABSORPTION AND CONSENSUS: Absorption: A + AB = A | A·(A+B) = A Consensus: AB + A'C + BC = AB + A'C
De Morgan's theorems are among the most important: they allow converting between AND-OR logic and NAND-NOR logic, enabling circuit optimisation across different gate families.
Real-World Applications
Apply boolean laws to reduce F = A'BC + AB'C + ABC' + ABC to simpler forms before building the circuit.
Fewer product terms in SOP form means fewer AND gates; simplification reduces total gate count and chip area.
Use De Morgan's theorems to convert AND-OR circuits to NAND-NAND implementations, halving transistor count.
Proving two circuits equivalent requires showing their boolean expressions are algebraically identical — formal hardware verification.
C/C++ compilers apply boolean identities to optimize conditional expressions and bitwise operations in generated code.
Try It in the Interactive Simulator
Build Boolean Algebra Rules and Laws circuits in real time — drag gates, connect wires, toggle inputs, and see outputs update instantly.
Frequently Asked Questions
- What are De Morgan's theorems?
De Morgan's theorems: (1) The complement of an AND is the OR of complements: (A·B)' = A' + B'. (2) The complement of an OR is the AND of complements: (A+B)' = A'·B'. They are essential for converting AND-OR logic to NAND/NOR implementations.
- What is the absorption law?
Absorption law: A + AB = A and A(A+B) = A. A term that already contains A is absorbed into A because it adds no new information. Example: X + XY = X regardless of Y.
- What is the consensus theorem?
The consensus theorem states: AB + A'C + BC = AB + A'C. The term BC is redundant because it is implied by the other two terms. Removing redundant terms simplifies expressions without changing the truth table.
- How do I simplify using boolean algebra?
Identify applicable laws: look for complement pairs (A·A'=0), idempotents (A+A=A), absorption (A+AB=A), and De Morgan applications. Work step by step, applying one rule at a time, and verify with a truth table.
- What is the duality principle in boolean algebra?
The duality principle states that every boolean identity has a dual obtained by swapping AND↔OR and 0↔1. If F = G is a theorem, then its dual is also a theorem. This halves the number of laws you need to memorize.