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\section*{Relations}  Let $A$ and $B$ be two sets and $A \times B $ their Cartesian Product, we define \textbf{Relation} a subset $\mathcal{R}$ of $A \times B $. \ \\  E.g.:  \begin{center}  $A=\lbrace0,1,2\rbrace$ \ \\  $B=\lbrace-1,1\rbrace$ \ \\  $A \times B = \lbrace(0,-1),(0,1),(1,-1),(1,1),(2,-1),(2,1) \rbrace$ \ \\  we take e.g.: $\mathcal{R}= \lbrace(0,1),(2,-1) \rbrace$   \end{center}  and if we look we could say that this is the set of the pairs whose sum is 1 and so:  \begin{center}  $\mathcal{R}= \lbrace (a,b) \in A\times B \mid a+b=1 \rbrace$  \end{center}  or we could say that $a\mathcal{R} b \Leftrightarrow a+b=1$.\\   If we take $B=A$ then $A\times A=A^{2}$ and the relation $ \mathcal{R}$ is a binary relation.  Some properties of binary relations:  \begin{itemize}  \item \textbf{Reflexive} $$\forall a \in A \ \ \ \ a \mathcal{R} a $$  \item \textbf{Symmetric} $$\forall a,b \in A \ \ if \ a \mathcal{R} b \Rightarrow b \mathcal{R} a $$  \item \textbf{Antisymmetric} $$\forall a,b \in A \ \ if \ a \mathcal{R} b \ and \ b \mathcal{R} a \Rightarrow a=b$$  \item \textbf{Transitive} $$\forall a,b,c \in A \ \ if \ a \ \mathcal{R} b \ and \ b \mathcal{R} c \Rightarrow a \mathcal{R} c$$  \end{itemize}  and so we could define   \begin{center}  \textbf{Equivalence Relation} if it is Reflexive,Symmetric and Transitive \ \\  \textbf{Order Relation} if it is Reflexive, Antisymmetric and Transitive.  \end{center}  \ \\  In particular, let $\mathcal{R}$ be an Equivalence Relation over the set $A$, we could regroup the elements that are related each other in the same class with an unique representative, i.e. $x$, and this is called \textit{equivalence class of the element $x$}, denoted by $[x]$. Finally we call \textbf{Quotient Set} :  \begin{center}  $A_{/\mathcal{R}}=\big\lbrace[x] \ \ | \ \ x \in A \big\rbrace$  \end{center}  The quotient set will play a main role in the construction of the cryptography over particular finite sets.