---
tag: 1MF0TVHDZD
kind: concept
name: Free Boolean algebra
msc: ["06E05", "03G05"]
specializes: [0XADJKM699]
sources:
  - url: https://en.wikipedia.org/wiki/Free_Boolean_algebra
same_as:
  wikidata: Q5499761
  wikipedia: Free Boolean algebra
---

## Summary

The free Boolean algebra on a set $S$ is the [[Boolean algebra|0XADJKM699]] generated by $S$ with no extra relations.
The only equalities in it are those that hold in every Boolean algebra.
Every map from $S$ into a Boolean algebra extends to exactly one homomorphism from it.
On $n$ generators it is the algebra of all [[Boolean functions|0X05GS525E]] of $n$ variables, with $2^{2^n}$ elements.
It is also the Lindenbaum–Tarski algebra of classical propositional logic.
Its propositional variables are the elements of $S$.

## Explanations

### L1 human

Take letters $x, y, z, \ldots$ and form all formulas with "and", "or" and "not".
Two formulas count as the same when they have the same truth table.
The resulting algebra is the free Boolean algebra on the letters.
With one letter $x$ it has four elements: false, $x$, not $x$, and true.