# Partial Order Relation

DEFINITION1: Let $X$ be a set and $R\subset{X\times{X}}$. If the relation $R$ is reflexive, antisymmetric and transitive, then the relation $R$ is called a "partial order relation" and denoted by $R=\le$ in general. If "$\le$" is a partial order relation over a set $X$, then $(X,\le)$ is called "partially ordered set" or shortly "poset".

DEFINITION2: Let $x$ and $y$ are elements of a partially ordered set $X$. If it holds “$x\le{y}\lor{y\le{x}}$”, then $x$ and $y$ are called “comparable”. Otherwise they are called “incomparable”.

DEFINITION3: If $x$ and $y$ are comparable for all $x,y$ in a partially ordered set $(X,\le)$, then the relation $\le$ is called a “total order” and the set $X$ is called a “totally ordered set” or “linearly ordered set”.

DEFINITION4: Let $(X,\le)$ be a partially ordered set and $A\subset{X}$. If $(A,\le)$ is a totally ordered set, then $A$ is called a “chain” in $X$.

DEFINITION5: Let $(X,\le)$ be a partially ordered set and $A\subset{X}$. If there exists an element $a^{*}\in{A}$ satisfying $a\le{a^{*}}$ for all $a\in{A}$, then $a^{*}$ is called the maximum of $A$, and if there exists an element $a_{*}\in{A}$ satisfying $a_{*}\le{a}$ for all $a\in{A}$, then $a_{*}$ is called the minimum of $A$. The minimum and the maximum of $A$ are denoted by $\min{A}$ and $\max{A}$ respectively.

# Relation

DEFINITION1: Let $X$ and $Y$ be two sets. Any subset of the cartesian product $X\times{Y}$ is called a relation with domain $X$ and codomain $Y$. While some sources are giving the definition of relation, they assume $X,Y\ne{\varnothing}$ and it’s said the emptyset being a subset of $X\times{Y}$ isn’t a relation. However, the assumption “the emptyset is a relation” is not a problem for any branch of the mathematics. On the contrary, the assumption “the emptyset is a relation” plays an important role in some branch of the mathematics.

If $X$ and $Y$ are two sets with $n$ and $m$ elements respectively, then the cartesian product $X\times{Y}$ has $n.m$ elements. Since a relation with domain $X$ and codomain $Y$ is an element of the power set $\mathbf{P}(X\times Y)$ and the number of the elements of the power set of a set with $k$ elements is $2^{k}$, then the number of all the relations with domain $X$ and codomain $Y$ is $2^{n.m}$. If $X,Y\ne{\varnothing}$ and at least one of $X$ and $Y$ is infinite set, then the number of all the relations with domain $X$ and codomain $Y$ is also infinity.

Let $R\subset{X\times{Y}}$ be a relation not being the emptyset. The statement $(x,y)\in{R}$ is read “x is R-related to y” and is denoted by $xRy$ or $R(x)=y$.

EXAMPLE1: Let $X=\{a,b,c\}$ and $Y=\{1,2\}$. Since $X$ has 3 elements and $Y$ has 2 elements, the number of all the relations with domain $X$ and codomain $Y$ is $2^{2.3}=2^{6}=64$. We can give some of these $64$ relations:

$R_1=\varnothing$,