# Strict ordering

## Definition

A binary relation $$\mathrel{R}$$, on a set $$S$$, is an ordering if and only if:

$$\mathrel{R}$$ is asymmetric:
$$\forall x,y \in S : x{\mathrel{R}}y \Rightarrow \neg(y{\mathrel{R}}x)$$
$$\mathrel{R}$$ is transitive:
$$\forall x,y,z \in S : x{\mathrel{R}}y \wedge y{\mathrel{R}}z \Rightarrow x{\mathrel{R}}z$$