Boolean Algebra

up:: Lattice

A Universal Algebra is said to be a Boolean Algebra if it satisfies the properties:

• is a Distributive and Limited Lattice ()
• To every point , there is a unique complement

It is assured to have unique complement, since Every distributive and limited lattice has unique complements.

Properties

De Morgan Laws

Every boolean algebra satisfies the De Morgan laws:

To prove the first equation¹:

Since every element in has unique complement, we have that .

Examples

• Measurable Spaces are examples of boolean algebras with elements from its Sigma-algebra.

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 02

Footnotes

1. The proof of the second equation follows from proving the first, by switching all elements , . ↩