---
tag: 06GKBTAA5E
kind: definition
name: Boolean algebra
msc: ["06E05"]
defines: 0XADJKM699
sources:
  - url: https://en.wikipedia.org/wiki/Boolean_algebra_(structure)
    locator: Definition
---

## Definition

A Boolean algebra is a set $A$ with two binary operations $\wedge$ and $\vee$, a unary operation $\neg$, and two elements $0, 1 \in A$.
These axioms must hold for all $a, b, c \in A$.
Associativity: $a \vee (b \vee c) = (a \vee b) \vee c$ and $a \wedge (b \wedge c) = (a \wedge b) \wedge c$.
Commutativity: $a \vee b = b \vee a$ and $a \wedge b = b \wedge a$.
Absorption: $a \vee (a \wedge b) = a$ and $a \wedge (a \vee b) = a$.
Identity: $a \vee 0 = a$ and $a \wedge 1 = a$.
Distributivity: $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$ and $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$.
Complements: $a \vee \neg a = 1$ and $a \wedge \neg a = 0$.

## Remarks

The relation $a \le b$, defined by $a \wedge b = a$, is a partial order with least element $0$ and greatest element $1$.
Equivalently, a Boolean algebra is a complemented [[distributive lattice|0T36P6TXQ6]].
The power set of a set is a Boolean algebra, the [[power set Boolean algebra|004J7ZEFAT]]. Boolean algebras correspond to [[Boolean rings|0N5FCV2YV6]].

## Explanations

### L1 human

A Boolean algebra has operations that behave like and, or and not on true and false, or like intersection, union and complement on sets.
Its axioms are the common rules of logic and of set algebra.