Relation

A set of Tuple showing how a set is related.

Formal Definition

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

• $\{(a,b)|aRb\}$

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

Example

$\{(x,y)|x\text{ loves }y\}$

Types

• Binary Relation
• Unique Relation
• Total Relation