# Orderings

## Ordering

### Inequalities

#### Less than or equal

Orderings define relations between elements in a set, where one element can preceed the other.

Orderings are antisymmetric. That is, the only case where the relation is satisfied in both directions is if the elements are equal.

\((a\le b)\land (b\le a)\rightarrow (a=b)\)

Orderings are transitive. That is:

\((a\le b)\land (b\le c)\rightarrow (a\le c)\)

#### Greater than or equal

#### Less than and greater than

The relation \(\le \) is refered to as non-strict.

There is a similar strict relation relation, \(<\):

\((a\le b)\land \neg (b\le a)\rightarrow (a < b)\)