Boolean Algebra

up:: Lattice

A Universal Algebra $(A,\{\wedge ,\vee ,\neg ,1,0\})$ is said to be a Boolean Algebra if it satisfies the properties:

• $(A,\wedge ,\vee )$ is a Distributive and Limited Lattice ($\forall x\in X:x\wedge 0=0,x\vee 1=1$)
• To every point $x\in X$, there is a unique complement $\neg x\in X$

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:

\begin{cases*} \lnot (a \land b) = \lnot a \lor \lnot b\\ \lnot(a \lor b) = \lnot a \land \lnot b \end{cases*}

To prove the first equation¹:

$$\begin{array}{rl}(a\wedge b)\wedge (\neg a\vee \neg b)& =a\wedge [(b\wedge \neg a)\vee (b\wedge \neg b)]\\ & =a\wedge (b\wedge \neg a)\\ & =(a\wedge \neg a)\wedge b\\ & =0\end{array}$$

Since every element in $X$ has unique complement, we have that $\neg (a\wedge b)=\neg a\vee \neg b$.

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 $a\to \neg a$, $b\to \neg b$. ↩