Relation

This is how elements of a set are related.

Formal Definition

A relation is a subset of Cartesian Product of two sets. $R\subset A\times B$

Properties

Reflexive

$R$ is reflexive if $\forall a\in A,(a,a)\in R$

Symmetric

$R$ is symmetric if $\forall a,b\in A,(a,b)\in R⟹(b,a)\in R$

Transitive

$R$ is transitive if $\forall a,b,c\in A,(a,b)\in \mathbb{R}\wedge (b,c)\in R⟹(a,c)\in R$

Antisymmetric

R is antisymmetric if $\forall a,b\in A$ s.t $a\ne b,(a,b)\in R⟹(b,a)\notin R$

Partial Order

$R$ is a partial order if it is Antisymmetric and Transitive