Boolean Algebra
Boolean algebra is a branch of algebra in which the variables can only take the values 0 (false) or 1 (true), and the operations are AND (conjunction, ·), OR (disjunction, +), and NOT (negation, ‾). It provides the mathematical framework for analysing and simplifying digital logic circuits, and is the theoretical foundation of computer science and digital electronics. Boolean expressions can be simplified using theorems and laws such as De Morgan's Theorems, the absorption law, and consensus theorem to reduce the number of logic gates required in a circuit.
Key Formula
NOT(A AND B) = NOT-A OR NOT-B (De Morgan's First Theorem)
LaTeX: \overline{A \cdot B} = \bar{A} + \bar{B} \quad \text{(De Morgan's First Theorem)}
| Symbol | Meaning | Unit |
|---|---|---|
| A, B | Boolean variables (can be 0 or 1) | dimensionless (binary) |
| \cdot | AND operation (logical conjunction) | Boolean operator |
| + | OR operation (logical disjunction) | Boolean operator |
| \bar{A} | NOT A (logical complement of A) | Boolean operator |
Worked Example
Problem
Simplify the Boolean expression Y = AB + AB̄ using Boolean algebra laws.
Solution
Step 1: Factor out A: Y = A(B + B̄). Step 2: Apply complement law: B + B̄ = 1. Step 3: Apply identity law: A × 1 = A. Step 4: Therefore Y = A. Step 5: Verification with truth table: A=0, B=0: Y = 0·0 + 0·1 = 0 = A ✓ A=0, B=1: Y = 0·1 + 0·0 = 0 = A ✓ A=1, B=0: Y = 1·0 + 1·1 = 1 = A ✓ A=1, B=1: Y = 1·1 + 1·0 = 1 = A ✓
Answer
Y = A (the expression simplifies to A)
Fundamental Laws and Theorems of Boolean Algebra
| Law / Theorem | AND Form | OR Form | Description |
|---|---|---|---|
| Identity Law | A · 1 = A | A + 0 = A | Operating with neutral element |
| Null Law | A · 0 = 0 | A + 1 = 1 | Domination by 0 or 1 |
| Idempotent Law | A · A = A | A + A = A | Repeating same variable |
| Complement Law | A · Ā = 0 | A + Ā = 1 | Variable with its complement |
| De Morgan's 1st Theorem | ̄(A · B) = Ā + B̄ | — | NAND → OR of complements |
| De Morgan's 2nd Theorem | — | ̄(A + B) = Ā · B̄ | NOR → AND of complements |
| Absorption Law | A · (A + B) = A | A + A · B = A | Simplification by absorption |
Interactive Tools
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Related Terms
Logic Gate
A logic gate is a fundamental electronic circuit that performs a basic Boolean logic operation on one or more binary inputs (0 or 1) to produce a single binary output, forming the building blocks of all digital systems. The primary logic gates are AND, OR, NOT, NAND, NOR, XOR, and XNOR, each implementing a specific logical function defined by a truth table. Logic gates are implemented using transistors and are combined in large numbers to create complex digital circuits such as processors, memory units, and controllers.
Transistor
A transistor is a semiconductor device with three terminals that can amplify electrical signals or act as an electronic switch by controlling current flow between two terminals using a small input signal at the third. The two main types are the Bipolar Junction Transistor (BJT), which is current-controlled, and the Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET), which is voltage-controlled. Transistors are the fundamental building blocks of modern electronics, with billions packed into a single integrated circuit chip.
Superposition Theorem
The Superposition Theorem states that in any linear circuit with multiple independent sources, the response (voltage or current) at any element equals the algebraic sum of the responses caused by each independent source acting alone, with all other independent sources deactivated. Voltage sources are deactivated by replacing them with short circuits, while current sources are deactivated by replacing them with open circuits. This theorem greatly simplifies the analysis of circuits with multiple sources and applies only to linear systems.
Boolean algebra is named after George Boole (1815–1864), an English mathematician and logician who introduced the system in his 1847 work "The Mathematical Analysis of Logic" and expanded it in "An Investigation of the Laws of Thought" (1854). "Algebra" comes from Arabic "al-jabr" (reunion of broken parts), introduced to Europe through al-Khwarizmi's 9th-century work.