$$n$$-ary relation
An $$n$$-ary relation on the sets $$X_{1} , X_{2} , \ldots , X_{n}$$ is a subset $$R$$ of their cartesian product, where we say that the relation holds between $$x_{1} , x_{2} , \ldots , x_{n}$$ when $$\left ( x_{1} , x_{2} , \ldots , x_{n} \right ) \in R$$
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 $$x R y$$ to say that $$x$$ is related to $$y$$.

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

set filtered by a relation
Suppose that $$X$$ is a set, and that $$R$$ is a unary relation on $$X$$, then we define $$X^{R} := \left \lbrace a \in X : a \in R \right \rbrace$$
reflexive relation
Given a binary relation $$R$$ on $$X$$, we say it is reflexive when every element of $$X$$ is related to itself. Symbolically $$\forall x \in X , x R x$$
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 $$x R y$$ then $$y R x$$
anti-symmetric relation
Given a binary relation $$R$$ on $$X$$, we say it is anti-symmetric when for any $$x , y \in X$$, if $$x R y$$ and $$y R x$$ we have that $$x = y$$
transitive relation
Given a binary relation $$R$$ on $$X$$, we say it is transitive if for any $$x , y , z \in X$$ if $$x R y$$ and $$y R z$$ then $$y R x$$
equivalence relation

An equivalence relation is a reflexive, symmetric and transitive relation

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

equivalence class
Given an equivalence relation $$~$$ an equivalence class is a complete set of equivalent elements, which we denote by $$\left [ x \right ]$$ which is defined to be the set $$\left \lbrace y \in X : y ~ x \right \rbrace$$
congruences induce a equivalence relation
Fix $$n \in \mathbb{N}_{1}$$, then relation on $$\mathbb{Z}$$ by $$a R b$$ if and only if $$a \% n = b \% n$$ defines an equivalence relation

Given any $$a \in G$$, then clearly $$a \% n = a \% n$$ so that $$a R a$$, if $$a R b$$, then $$a \% n = b \% n$$ and then $$b \% n = a \% n$$, finally if $$a \% n = b \% n$$ and $$b \% n = c \% n$$ therefore $$a \% n = c \% n$$

partial order

a partial order is a binary relation $$R$$ on a set $$X$$ such that for any $$a , b , c \in X$$ we have that:

• $$a R a$$
• if $$a R b$$ and $$b R a$$ then $$a = b$$
• if $$a R b$$ and $$b R c$$ then $$a R c$$
And we say that $$X$$ is a partially ordered set.

Note that the above three statements equivalently say that $$\le$$ is a reflexive, anti-symmetric and transitive relation.

total order
A total order is a partial order
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