attachment:booleanalgebra.tex of BooleanAlgebra

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  10 %TCIDATA{Created=Sunday, January 23, 2005 12:05:04}
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  27 \newtheorem{definition}[theorem]{Definition}
  28 \newtheorem{example}[theorem]{Example}
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  41 \begin{document}
  42 
  43 
  44 \section{Boolean Algebra Definitions and Examples}
  45 
  46 \begin{definition}
  47 A \textbf{Boolean Algebra} is a tuple%
  48 \[
  49 \left\langle \delta ,\sqcap ,\sqcup ,^{\prime },\top ,\bot \right\rangle 
  50 \]%
  51 Where each element is defined as follows: 
  52 \[
  53 \begin{tabular}{ll}
  54 $\delta $ & Non-empty set called the \textbf{domain} \\ 
  55 $\sqcap $ & binary function $\sqcap :\delta \times \delta \rightarrow \delta 
  56 $ \\ 
  57 $\sqcup $ & binary function $\sqcup :\delta \times \delta \rightarrow \delta 
  58 $ \\ 
  59 $^{\prime }$ & unary function $^{\prime }:\delta \rightarrow \delta $ \\ 
  60 $\bot $ & specific element called the \textbf{zero }element \\ 
  61 $\top $ & specific element called the \textbf{one} element%
  62 \end{tabular}%
  63 \]
  64 \end{definition}
  65 
  66 \begin{definition}
  67 A Boolean algebra of sets \emph{is any }\ Boolean algebra, where:
  68 
  69 $\delta $ is a set of sets,
  70 
  71 $\sqcup $ is interpreted as set union, denoted $\cup $.
  72 
  73 $\sqcap $ is interpreted as set intersection denoted $\cap $
  74 
  75 $^{\prime }$ is interpreted as set complement with repsect to $\top $,
  76 denoted $\overline{}$, and $\sqsubseteq $ (or $\sqsupseteq $) is interpreted
  77 as set containtment, denoted as $\subseteq $ (or $\supseteq $ ).
  78 \end{definition}
  79 
  80 \begin{definition}
  81 An \emph{atom} of a Boolean algebra is an element $x\neq \bot $ such that
  82 there is no other element $y\neq \bot $ with $y\sqsubseteq x$. I can happen
  83 that there are no atoms at all in a Boolean algebra. In that case we call
  84 the Boolean algebra \emph{atomless}; otherwise we call it \emph{atomic}.
  85 \end{definition}
  86 
  87 \begin{example}
  88 \[
  89 B_{Z}=\left\langle Powerset(\mathbb{Z}),\cap ,\cup ,~%
  90 %TCIMACRO{\U{af}}%
  91 %BeginExpansion
  92 \bar{}%
  93 %EndExpansion
  94 ~,\emptyset ,\mathbb{Z}\right\rangle 
  95 \]%
  96 is a Boolean algebra of sets. In this algebra for each $i\in \mathbb{Z}$,
  97 the singleton $\left\{ i\right\} $ is an atom. Thus this is an \textbf{%
  98 atomic Boolean algebra}. Clearly $x=\left\{ 1\right\} \neq \bot $ and $y$
  99 can be either $\left\{ 1\right\} $ or $\emptyset $. We eliminate $\left\{
 100 1\right\} $ from consideration because it is not an \textbf{other element}.
 101 Since $y=\bot $, we conclude that there are no other elements such that $%
 102 y\neq \bot $.
 103 \end{example}
 104 
 105 \begin{example}
 106 Let $H$ be the set of all finite unions of half-open intervals of the form $%
 107 [a,b)$ over the rational numbers, where $[a,b)$ means all rational numbers
 108 that are greater than or equal to $a$ and less than $b$, where $a$ is a
 109 rational or $-\infty $ and $b$ is a rational number or $\infty $.%
 110 \[
 111 B_{H}=\left\langle H,\cap ,\cup ,~%
 112 %TCIMACRO{\U{af}}%
 113 %BeginExpansion
 114 \bar{}%
 115 %EndExpansion
 116 ~,\emptyset ,\mathbb{Q}\right\rangle 
 117 \]%
 118 This is another Boolean algebra of sets, but this set is atomless. $\forall
 119 x=[a,b)$, $a<b$ there exists a $c$ such that $a<c<b$. Thus $[a,c)\subseteq
 120 \lbrack a,b)$, and $[a,c)\neq \emptyset $.
 121 \end{example}
 122 
 123 \end{document}