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.

[external website]

Author: Henry Cohn

-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