5.4 Relations

1. Let $A$ and $B$ be sets. A $\text{relation}$ from $A$ to $B$ is a subset $A\times B$
2. Let $k\in {\mathbb{Z}}^{+}$ and let $A_1,\dots ,A_k$ be sets
   A $\text{relation on }{A}_1,\dots ,A_k$ is a subset of $A_1\times \cdots \times A_k$
   The number $k$ is called the $\text{arity}$ of the relation 3. Let $A$ be a set and let $k\in \mathbb{N}$
   A $k\text{-ary relation over }A$ is a subset of $A^k$

#TODO the example

Let $A$ be a set. A $\text{relation on the set }A$ is a relation from $A$ to itself.

#TODO Continue example

A relation $R$ on a set $A$ is called $\text{reflexive}$ if $(a,a)\in R$ for every element $a\in A$

A relation $R$ on a set $A$ is called $\text{symmetric}$, if $(a,b)\in R$ implies $(b,a)\in R$ for all $a,b\in A$

A relation $R$ on a set $A$ such that for all $a,b\in A$ if $(a,b)\in R$ and $(b,a)\in R$, then $a=b$, is called $\text{antisymmetric}$

A relation $R$ on a set $A$ is called $\text{transitive}$ if whenever $(a,b)\in R$ and $(b,c)\in R$, then $(a,c)\in R$, for all $a,b,c\in A$

A relation $R$ on a set $A$ is called $\text{total}$ if all $a,b\in A$ satisfy: $(a,b)\in R$ or $(b,a)\in R$

An $\text{equivalence relation}$ is a binary relation, that is reflexive, transitive and symmetric

Let $E$ be an equivalence relation over a set $V$.
For every $v\in V$ we let$[v{]}_E:=\{v^{\prime }\in V|(v,v^{\prime })\in E\}$denote the $\text{equivalence class of}$ $v$ with respect to $E$ $\text{(i.e. the equivalence class }[V{]}_E\text{ consists of all elements of }V$
$\text{that are }\text{‘}\text{equivalent}\text{’}\text{ to }v\text{ according to }E\text{).}$
A set $W\subseteq V$ is an $\text{equivalence class}$ (of $E$), if there exists an element $v\in V$ with $W=[v{]}_E$
The element $v$ is then called a $\text{representative}$ of its equivalence class $W$