---
tag: 0WG43AC0EV
kind: example
name: Two-element Boolean algebra
msc: ["06E05"]
instance_of: [0XADJKM699]
sources:
  - url: https://en.wikipedia.org/wiki/Two-element_Boolean_algebra
    locator: Definition
same_as:
  wikidata: Q7858706
  wikipedia: Two-element Boolean algebra
---

## Example

The two-element Boolean algebra is the set $\{0, 1\}$ with these operations.
$a \wedge b = 1$ if and only if $a = b = 1$.
$a \vee b = 0$ if and only if $a = b = 0$.
$\neg 0 = 1$ and $\neg 1 = 0$.
It satisfies the axioms of a [[Boolean algebra|06GKBTAA5E]], with $0$ as least element and $1$ as greatest element.

## Remarks

If $1$ means true and $0$ means false, then $\wedge$, $\vee$ and $\neg$ are the connectives and, or and not.
An equation in the Boolean operations holds in all Boolean algebras if and only if it holds in the two-element Boolean algebra.
It is isomorphic to the power set of a set with one element.

## Explanations

### L1 human

The two-element Boolean algebra is the algebra of true and false.
It is also the algebra of one bit in a digital circuit.