$n$-ary relation
An $n$-ary relation on the sets ${X}_{1},{X}_{2},\dots ,{X}_{n}$ is a subset $R$ of their cartesian product, where we say that the relation holds between ${x}_{1},{x}_{2},\dots ,{x}_{n}$ when $\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)\in R$
Unary Relation
A unary relation $R$ on X is a $1$-ary relation
Binary Relation

A binary relation $R$ is a $2$-ary relation, when $\left(x,y\right)\in R$, then we write $xRy$ to say that $x$ is related to $y$.

Note: A binary relation on $X×X$ is stated simply as a relation on $X$

The Reverse of a Binary Relation
Suppose that $R$ is a binary relation then $\mathrm{rev}\left(R\right):=\left\{\left(y,x\right):\left(x,y\right)\in R\right\}$
set filtered by a relation
Suppose that $X$ is a set, and that $R$ is a unary relation on $X$, then we define ${X}^{R}:=\left\{a\in X:a\in R\right\}$
reflexive relation
Given a binary relation $R$ on $X$, we say it is reflexive when every element of $X$ is related to itself. Symbolically $\mathrm{\forall }x\in X,xRx$
symmetric relation
Given a binary relation $R$ on $X$, we say it is symmetric if the relation goes both ways, that is for any $x,y\in X$ if $xRy$ then $yRx$
Anti-Symmetric Relation
Given a binary relation $R$ on $X$, we say it is anti-symmetric when for any $x,y\in X$, if $xRy$ and $yRx$ we have that $x=y$
Asymmetric Relation
Given a binary relation $R$ on $X$, we say it is asymmetric when for any $x,y\in X$, if $xRy$ then it's false that $yRx$.
Transitive Relation
Given a binary relation $R$ on $X$, we say it is transitive if for any $x,y,z\in X$ if $xRy$ and $yRz$ then $yRx$
Equivalence Relation

An equivalence relation is a reflexive, symmetric and transitive relation

when $R$ is an equivalence relation, we denote it by $\sim$ instead of $R$

Connected Relation
Given a binary relation $R$ on $X$, we say it is connected if for any $x,y\in X$ if $x\ne y$ then $xRy$ or $yRx$
Strongly Connected Relation
Given a binary relation $R$ on $X$, we say it is strongly connected if for any $x,y\in X$ $xRy$ or $yRx$
Equivalence Class
Given an equivalence relation $\sim$ an equivalence class is a complete set of equivalent elements, which we denote by $\left[x\right]$ which is defined to be the set $\left\{y\in X:y\sim x\right\}$
Congruences induce a Equivalence Relation
Fix $n\in {\mathbb{N}}_{1}$, then relation on $\mathbb{Z}$ by $aRb$ if and only if $a\mathrm{%}n=b\mathrm{%}n$ defines an equivalence relation

Given any $a\in G$, then clearly $a\mathrm{%}n=a\mathrm{%}n$ so that $aRa$, if $aRb$, then $a\mathrm{%}n=b\mathrm{%}n$ and then $b\mathrm{%}n=a\mathrm{%}n$, finally if $a\mathrm{%}n=b\mathrm{%}n$ and $b\mathrm{%}n=c\mathrm{%}n$ therefore $a\mathrm{%}n=c\mathrm{%}n$

Partial Order

A partial order is a binary relation denoted by $\le$ on a set $X$ such that $\le$ is reflexive, anti-symmetric and transitive relation.

Strict Partial Order
A strict partial order is a binary relation denoted by $<$ on a set $X$ such that $<$ is transitive and asymmetric.
Every Partial Order Admits a Strict Partial Order
Suppose that $\left(X,\le \right)$ is a partial order, then there exists a relation $<$ such that $\left(X,<\right)$ is a partial order such that $a
Consider the partial order $\left(X,\le \right)$, since $\le$ is a collection of tuples we just need to consider $<=\le \setminus \left\{\left(x,x\right):x\in X\right\}$
Comparable
Suppose $R$ is a partial order, then elements $a,b\in R$ are said to be comparable if $aRb$ or $bRa$, otherwise they are said to be incomparable

With the strict partial order $\subset$ we can see that $\left\{1\right\}$ and $\left\{2\right\}$ are incomparable.

Total Order
A total order is a partial order which is strongly connected.
well order
A well order on a set $S$ is a total order on $S$ along with the property that every non-empty subset of $S$ has a least element under this ordering
well ordering
Every set $X$ can be well ordered
TODO
The Smallest Element of a Set is Unique
Supose that $\left(S,\le \right)$ is a partial order, then the smallest element is unique
Suppose that there are two smallest elements $a,b\in S$ since $a$ is the smallest then $a\le b$ and since $b$ is the smallest then $b\le a$ by transitivity $a=b$ and so the smallest element is unique.