Boolean Algebra: Logic and Binary Variables
Explore Boolean algebra, a mathematical system dealing with binary variables and logical operations (AND, OR, NOT). This guide explains the axioms of Boolean algebra, its key properties, and its applications in digital logic and computer science.
Boolean Algebra
What is Boolean Algebra?
Boolean algebra is a type of algebra dealing with binary variables (variables that can only have two values, typically 0 and 1, representing false and true). It uses logical operations (AND, OR, NOT) to manipulate these variables. A Boolean algebra is formally defined as a complemented distributive lattice.
Boolean Algebra: Definition
A Boolean algebra is a set B with operations AND (∧), OR (∨), and NOT (') and special elements 0 and 1, satisfying specific axioms. These axioms ensure that Boolean algebra follows consistent rules.
Properties of Boolean Algebra
Boolean algebra follows several key properties:
Commutative Properties
- a + b = b + a
- a ∧ b = b ∧ a
Distributive Properties
- a + (b ∧ c) = (a + b) ∧ (a + c)
- a ∧ (b + c) = (a ∧ b) + (a ∧ c)
Identity Properties
- a + 0 = a
- a ∧ 1 = a
Complement Laws
- a + a' = 1
- a ∧ a' = 0
(Where a' represents the complement of a.)
Sub-Algebras
A sub-algebra of a Boolean algebra B is a subset of B that is itself a Boolean algebra under the same operations. It must contain 0 and 1 and be closed under all three operations (AND, OR, NOT).
Example: Sub-Algebra
(This example, showing sub-algebras of D₇₀, is given in the original text and should be included here.)
Isomorphic Boolean Algebras
Two Boolean algebras B and B₁ are isomorphic if there's a one-to-one correspondence (bijection) between their elements that preserves the Boolean operations (AND, OR, NOT). This means they have the same structure, even if the elements themselves are different.
Example: Isomorphic Boolean Algebras
(This example, showing an isomorphism between a power set and a Boolean algebra defined on a set with a prime number, is given in the original text and should be included here.)
Basic Properties of Boolean Algebra
| Property | Formula |
|---|---|
| Ordering | a ≤ b if and only if a + b = b |
| Idempotent Laws | a + a = a; a ∧ a = a |
| Commutative Property | a + b = b + a; a ∧ b = b ∧ a |
| Absorption Laws | a + (a ∧ b) = a; a ∧ (a + b) = a |
| Identity Laws | a + 0 = a; a ∧ 1 = a |
| Null Laws | a ∧ 0 = 0; a + 1 = 1 |
| Distributive Laws | a ∧ (b + c) = (a ∧ b) + (a ∧ c); a + (b ∧ c) = (a + b) ∧ (a + c) |
| Complement Laws | 0' = 1; 1' = 0; a + a' = 1; a ∧ a' = 0 |
| Involution Law | (a')' = a |
| De Morgan's Laws | (a ∧ b)' = a' + b'; (a + b)' = a' ∧ b' |
Boolean Functions
A Boolean function maps n Boolean inputs to a single Boolean output. It can be represented using a truth table or a Boolean expression.
Examples: Boolean Functions
(Two examples of Boolean functions, one mapping {0,1}³ to {0,1} and another mapping {0,1,2,3}² to {0,1,2,3}, are shown in the original text using truth tables and should be included here.)
Conclusion
Boolean algebra is a fundamental tool in computer science and digital logic, providing a framework for analyzing and designing digital circuits and systems.