deletions | additions       diff --git a/Graduation Thesis.tex b/Graduation Thesis.tex  index 5f69248..bb764a2 100644  --- a/Graduation Thesis.tex  +++ b/Graduation Thesis.tex 
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\item The $\cdot$ is distributive with respect to the $\star$, left and right :  $$\forall a,b,c \in A \ \ \ \ a \cdot (b \star c )= a \cdot b \star a \cdot c $$  $$\forall a,b,c \in A \ \ \ \ (b \star c ) \cdot a = b \cdot a \star c \cdot a $$   \end{itemize}  We will call the unit of $\star$ as $0_{A}$, and the unit of $\cdot$ as $1_{A}$ 
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It could happen that two non-zero elements, when "multiplied", give as result $0_{A}$, in other words:  $$\exists a,b \in A \ \ \ \ \ a,b \neq 0_{A} \ \ \ s. \ t. \ a \cdot b= 0_{A} $$  These elements are called \textbf{Zero Divisors}.  \subsubsection{$\dagger$ Fields }  Let $F$ be a set. We define over $F$ the two previous operations $\star$ and $\cdot$; we call $(F,\star,\cdot)$ a \textbf{Field} if:  \begin{itemize}  \item $F$ is an Integral Domain  \item Every non-zero element of $F$ has multiplication inverse  \end{itemize}  \subsection{Number Sets: the Properties of $\mathbb{Z}$}  \subsubsection{$\dagger$ Number Sets}  The numbers that we know are divided in sets that are one included in the other:  \begin{itemize}  \item \textbf{Natural Numbers} $\mathbb{N}$ \\  The natural numbers are the ones given by the "nature", one apple, two trees, etc... they are all non negative.  $$\mathbb{N}=\lbrace1,2,3,4, \ldots \rbrace $$   \item \textbf{Integer Numbers} $\mathbb{Z}$ \\  The positive numbers united with 0 and the negative numbers (defined as the natural numbers with the minus beside).  $$\mathbb{Z}=\lbrace-2,-1,0,1,2, \ldots \rbrace $$  \item \textbf{Rational Numbers} $\mathbb{Q} $ \\  The Rational Numbers are the ones that are expressible by fraction.  $$\mathbb{Q}=\Bigg\lbrace -\dfrac{2}{3},0, 1, \dfrac{5}{3}, \ldots \Bigg\rbrace $$  \item \textbf{Real Numbers} $\mathbb{R} $ \\  All the numbers of our reality, rational and irrational.  $$\mathbb{R}=\Bigg\lbrace -\dfrac{2}{3},0, \sqrt{2}, \dfrac{5}{3}, \pi, \ldots \Bigg\rbrace $$  \item \textbf{Complex Numbers} $\mathbb{C}$ \\  The numbers that are constructed starting by assuming $\sqrt{-1}=i$ the \textbf{imaginary unit}. They have Real Part and Imaginary Part.  $$\mathbb{C}=\Bigg\lbrace -i,0,i,1+i,2+3i \ldots \Bigg\rbrace $$  \end{itemize}  As we immediately see the relation between these sets is:  $$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} $$  The Cryptography and the Cryptocurrencies, in some way, use the properties of the integers to found their basis.  \subsubsection{$\dagger$ The Set of Integers $\mathbb{Z}$}  We would recall some properties of the integers as the division, the existence of greatest common divisor and so on.  Firstly it's easy to prove that $(\mathbb{Z},+,\cdot)$ is a Ring, in particular is an Integral Domain. It's not a Field because e.g.  doesn't exist an integer $x \in \mathbb{Z}$ such that $2x=1$.