deletions | additions       diff --git a/bits1.tex b/bits1.tex  index e0dd757..d589e45 100644  --- a/bits1.tex  +++ b/bits1.tex 
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most of Computer Science applications.  \\  They are represented with the constants $0 $ and $1$ (the two events with the   same probability). The In Mathematics, the  set of the bits, in Mathematics bits  is usually denoted by %We have the Field of Bits $\Fb$. When we work in Computer Science we often   %consider 
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\caption{Sum and product}\label{SumProd}  \end{table}    The first and second columns represent the values of the input variables $a$ and $b$, the third and the fourth ones  respectively represent  the results of the sum and product of $a$ and $b$. $b$.\\  Performing operations with bits is not the same as doing so with ``usual''   (real) numbers (from now on, we denote the set of real numbers with $\RR$).  \\  Look now  at third row. We observe that that,  summing the bit $1$ with the bit $1$ $1$,  one gets $0$: $1+1=0$. $$1+1=0.$$  This also highlights that performing operations with bits is not the same as doing so with ``usual''   (real) numbers.  % indexed  
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% performing operations with bits is not the same as doing so with ``usual''   % (real) numbers.  Anyway, Even if the operations on $\Fb$ are similar but not equals to the corresponding ones on bits,  $\Fb$ has very good properties with respect to the operations $+$ and $\cdot$, that are in common with the real numbers, from now on represented with $\RR$. Indeed, we numbers.   \\  We  can observe that $\Fb$ is a \emph{field}, as well as $\RR$ is. \\  In particular particular,  every element of $\Fb$ has an opposite \emph{opposite}  for the sum, ``+'', ``$+$'',  and every element of $\Fb$, that is different from $0$, has an inverse, \emph{inverse},  exactly as the field of real numbers. For example in the field of real numbers \[  5+(-5)=0 \mbox{ and } 5\cdot \frac{1}{5}=1.   \]  The notation for In  the setof bits (that - from now on - we will call the \emph{field of bits}), reflects this fact. Indeed,  $\Fb$ stands for ``field'' and the subscript $2$ stands for the \emph{size} of the set itself, i.e. the number of its elements. That is we have:  \begin{enumerate}  \item The opposite of $1$ is $1$, in fact $1+1=0$.  \item The opposite of $0$ is $0$, as in $\RR$.  \item The inverse of $1$ is $1$, in fact $1\cdot 1=1$.  \item The inverse of $0$ does not exist as in $\RR$.  \end{enumerate}  The notation for the set of bits (that - from now on - we will call the \emph{field of bits}), reflects this fact. Indeed, $\mathbb{F}$ stands for ``field'' and the subscript $2$ stands for the \emph{size} of the set itself, i.e. the number of its elements.  %