# Boolean algebras

## Summary

Boolean algebras are a mathematical structure which shares the algebraic properties of propositional formulas. Canonical examples include propositional formulas and the power set of a set (with set union, intersection, and complement playing the roles of the propositional connectives). Boolean algebras are used in topology, model theory, and social choice theory.

## Context

This concept has the prerequisites:

- propositional logic (Boolean algebras are a generalization of propositional logic.)
- order relations (Boolean algebras are defined in terms of order relations.)
- set operations (Power sets, with the standard set operations, are a canonical example of a Boolean algebra.)
- countable sets (Some of the basic constructions of Boolean algebras involve countable sets.)

## Goals

- Define a lattice

- Define a Boolean algebra (as a complemented distributed lattice)

- Be able to prove simple facts about Boolean algebras

- Show that the power set of a set, with the standard set operations, forms a Boolean algebra

- Give an example of a Boolean algebra which is not a power set

- Define an atom

- Be aware of the principle of duality (that Boolean algebras are symmetric with respect to meet and join)

## Core resources (read/watch one of the following)

## -Free-

→ Notes on Logic (2013)

Lecture notes for a course on first order logic.

## -Paid-

→ A Course in Mathematical Logic

A graduate textbook in mathematical logic.

- Section 4.1, "Lattices," pages 125-129
- Section 4.2, "Boolean algebras," pages 129-132

Additional dependencies:

- topology of R^n
- first-order logic

## See also

-No Additional Notes-