# Equivalence Relation

DEFINITION1: Let $X$ be a set and $R\subset{X\times{X}}$. If the relation $R$ is reflexive, symmetric and transitive, then the relation $R$ is called an "equivalence relation" and denoted by $R=\sim$ in general.

DEFINITION2: Let $\sim$ be an equivalence relation over a set $X$ and $a\in{X}$. The set defined as $\{x\in{X}\:|\:a\sim{x}\}\subset{X}$ is called the “equivalence class” of $a$ under $\sim$ and denoted by $\overline{a}$, $[a]$ or $[a]_{\sim}$. Since $a\sim{a}$ for every $a\in{X}$, then $a\in{\overline{a}}$. So the equivalence class $\overline{a}$ is non-empty for every $a\in{X}$. The family of all the equivalence classes of the relation $\sim$ is called the “quotient set” of $X$ by $\sim$ and denoted by ${^X}/{_\sim}$

I.e.,

${^X}/{_\sim}=\{\overline{a}\:|\:a\in{X}\}\subset{\mathbf{P}(X)}$.

DEFINITION3: Let $\sim$ be an equivalence relation over a set $X$ and $a,b\in{X}$. If $b\in{\overline{a}}$, then $b$ is called a “representative class” of the equivalence class $\overline{a}$.

# 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.