Relations

Reflexive

The relation R on a set A is reflexive if a ordered pair (a,a) $\in$ R for every a $\in$ A $$\forall a \in A((a,a) \in R)$$

Symmetric

The relation R on a set A is symmetric if the ordered pairs (a,b) $\in$ R then (b,a) $\in$ R $$\forall a \forall b \left((a,b)\in R \rightarrow (b,a)\in R \right)$$

Anti-Symmetric

The relation R on a set A is antisymmetric if the ordered pairs (a,b) $\in$ R and (b,a) $\in$ R then a = b $$\forall a \forall b \left(( (a,b)\in R \wedge (b,a)\in R \rightarrow (a = b) \right)$$

Transative

The relation R on a set A is transative if the ordered pairs (a,b) $\in$ R and (b,c) $\in$ R then (a,c) $\in$ R $$\forall a \forall b \forall c \left(( (a,b)\in R \wedge (b,c)\in R \rightarrow (a,c) \in R \right)$$