# 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:

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

#### Less than and greater than

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

There is a similar strict relation relation, $$<$$:

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