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\section{Relations}

\begin{definition}
  Let $A$ be a set. If $R \subseteq A \times A$, then $R$ is a \emph{relation} on $A$. If $a,b\in A$, we denote $(a,b)\in R$ by $aRb$.
\end{definition}

\begin{definition}
  Let $R$ be a relation on a set $A$.
  \begin{enumerate}
    \item We call $R$ \emph{reflexive} if
    $$
      \forall a\in A \quad aRa
    $$
    \item We call $R$ \emph{irreflexive} if 
    $$
      \forall a\in A \quad \lnot aRa
    $$
    \item We call $R$ \emph{symmetric} if
    $$
      \forall a,b\in A \quad aRb \implies bRa
    $$
    \item We call $R$ \emph{asymmetric} if
    $$
      \forall a,b\in A \quad aRb \implies \lnot bRa
    $$
    \item We call $R$ \emph{antisymmetric} if
    $$
      \forall a,b\in A \quad aRb \land bRa \implies a=b
    $$
    \item We call $R$ \emph{transitive} if
    $$
      \forall a,b,c\in A \quad aRb \land bRc \implies aRc
    $$
    \item We call $R$ \emph{connected} if
    $$
      \forall a,b\in A \quad a\neq b \implies aRb \lor bRa
    $$
    \item We call $R$ \emph{strongly connected} if
    $$
      \forall a,b\in A \quad aRb \lor bRa
    $$
  \end{enumerate}
\end{definition}

\subsection{Equivalence Relations}

\begin{definition}
  A relation $R$ on a set $A$ is an \emph{equivalence relation} if $R$ is reflexive, symmetric, and transitive.
\end{definition}

\begin{definition}
  If $\sim$ is an equivalence relation on a set $A$, for each $a\in A$, the \emph{equivalence class} of $a$ is the set
  $$
    [a] = \{x\in A: x\sim a\}
  $$
\end{definition}

\begin{definition}
  A collection $\mathcal{A}$ of nonempty subsets of a set $A$ is a \emph{partition} of $A$ if $\cup\mathcal{A} = A$ and
  $$
    \forall X,Y\in\mathcal{A} \quad X\neq Y \implies X\cap Y = \emptyset
  $$
\end{definition}

\begin{theorem}
  If $\sim$ is an equivalence relation on a set $A$, then the set of equivalence classes is a partition of $A$.
  \begin{proof}
    Let $\sim$ be an equivalence relation on a set $A$. Let $\mathcal{A}$ be the set of equivalence classes. That is
    $$
      \mathcal{A} = \{[a] \in \mathcal{P}(A) : a\in A\}.
    $$
    Let $a\in A$. Since $\sim$ is reflexive, we have $a\sim a$. Then $a\in [a]$. Hence $A\subseteq \cup\mathcal{A}$.
    Since each equivalence class is a set of elements of $A$, we have $\cup\mathcal{A} \subseteq A$. Then $A = \cup\mathcal{A}$.
    Let $X,Y\in\mathcal{A}$. Then there are $x,y\in A$ such that $X=[x]$ and $Y=[y]$. Suppose $X \cap Y \neq \emptyset$.
    Then there is $z\in X\cap Y$. Then $z\in X$ and $z\in Y$. Hence $z\sim x$ and $z\sim y$. For any $u\in X$, we use symmetry and transitivity of $\sim$ to show that $u\sim x\sim z\sim y$, hence $u\in Y$ and $X\subseteq Y$. Likewise $Y\subseteq X$ and $X=Y$. It follows that $\mathcal{A}$ is a partition of $A$.
  \end{proof}
\end{theorem}
    
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