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% specific commands for boolfunc essay
\newcommand { \B } {\ensuremath{\mathbb{B}}}
\newcommand{\boolspace}[1][n]{\B^{#1}}
\newcommand{\exes}[1][n]{x_{1}, \dots, x_{#1}}
\newcommand{\pexes}[1][n]{\left(\exes[#1]\right)}
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\begin{document}

\title{Example}
\author{Anno Siegel}
\maketitle

\section{Boolean Spaces}\label{S:boolspace}

\begin{definition}\label{D:boolfield}Boolean Field\\
        Let $(\B, +, \cdot)$ denote the field $k_{2}$ with the usual
        field operations of integers modulo~2.

        We shall use two more operations on $\B$:

        $\lor$ (\emph{disjunction}) a binary operation,
        defined through $x \lor y = x + y + x y$, and \\
        $\lnot$ (\emph{negation}), a unary operation,
        defined through $\lnot x = 1 - x$, \\
        which make $(\B, \lor, \cdot, \lnot)$ a \emph{boolean algebra}.

    Negation can also be denoted by \emph{overlining}, so
    $\lnot x = \overline{x}$.
    
    We shall call $\B$ the \emph{boolean field}.
    
\end{definition}

That's a lot of notation for so small a field.  Here is an overview:\\
\begin{center}
\begin{tabular}{l|ll|ll}
    & \multicolumn{2}{c|}{Logical Term} & \multicolumn{2}{c}{Algebraic Term} \\
    \hline
    $+$ & exclusive or & xor & addition & plus \\
    $\cdot$ & conjunction & and & multiplication & times \\
    $\lor$ & disjunction & or & & \\
    $\lnot$ $\overline{\centerdot}$ & negation & not & &
\end{tabular}
\end{center}

The binary operations $+$, $\cdot$ and $\lor$ are associative and commutative.
Further, these distributive laws hold for all $x, y, z \in \B$:
\begin{align*}
    x(y + z) &= x y + x z \\
    x(y \lor z) &= x y \lor x z \\
    x \lor y z &= (x \lor y)(x \lor z)
\end{align*}

These are known as De Morgan's laws:
\begin{align*}
    \lnot(x y) &= \overline{x} \lor \overline{y} \\
    \lnot(x \lor y) &= \overline{x} \overline{y}
\end{align*}

Note that these rules of arithmetic (which we won't prove here) could
be derived directly from the definitions and the field properties of
$(\B, +, \cdot)$.

\begin{definition}\label{D:boolspace} Boolean Spaces\\
    Any vector space $V$ over $\B$ will be called a \emph{boolean space}.
    We shall also consider a boolean space a \emph{ring} with respect
    to the multiplication induced by the field multiplication.  Negation
    and disjunction also induce corresponding operations on $V$ which
    thus becomes a \emph{boolean algebra}.
\end{definition}

In this essay we shall only be concerned with finite-dimensional
boolean spaces mostly of the form $\B^{n}$.  These are the mathematical
model of what is aptly known as a \emph{bit vector} in computer lingo.
As usual, we shall represent $\vec{x} \in \B^{n}$ as
$\vec{x} = \pexes \text{ where } x_{i} \in \B$.

\end{document}