Equivalence of Definitions of Ordering/Proof 2

Theorem
The following definitions of ordering are equivalent:

Definition 1 implies Definition 2
Let $\RR$ be a relation on $S$ satisfying:

Condition $(1)$
Let $\tuple {x, y} \in \RR \circ \RR$.

Then there exists a $z \in \RR$ such that:


 * $\tuple {x, z}, \tuple {z, y} \in \RR$

By $\RR$ being transitive:


 * $\tuple {x, y} \in \RR$

Hence:


 * $\RR \circ \RR \subseteq \RR$

Now let $\tuple {x, y} \in \RR$.

By $\RR$ being reflexive:


 * $\tuple {y, y} \in \RR$

Hence by the definition of relation composition:


 * $\tuple {x, y} \in \RR \circ \RR$

Hence:


 * $\RR \subseteq \RR \circ \RR$

Condition $(2)$
Follows immediately from Relation is Antisymmetric iff Intersection with Inverse is Coreflexive and $\RR$ being reflexive.

Thus $\RR$ is an ordering by definition 2.

Definition 2 implies Definition 1
Let $\RR$ be a relation which fulfils the conditions:
 * $(1): \quad \RR \circ \RR = \RR$
 * $(2): \quad \RR \cap \RR^{-1} = \Delta_S$

Reflexivity
By Intersection is Subset the condition:


 * $\RR \cap \RR^{-1} = \Delta_S$

implies:


 * $\Delta_S \subseteq \RR$

Thus $\RR$ is reflexive by definition.

Antisymmetry
By Relation is Antisymmetric iff Intersection with Inverse is Coreflexive the condition:


 * $\RR \cap \RR^{-1} = \Delta_S$

implies that $\RR$ is antisymmetric.

Transitivity
Let $\tuple {x, y}, \tuple {y, z} \in \RR$.

Then by the definition of relation composition:


 * $\tuple {x, z} \in \RR \circ \RR$

But by the condition:


 * $\RR \circ \RR = \RR$

It follows that:


 * $\tuple {x, z} \in \RR$

Hence $\RR$ is transitive.

Thus $\RR$ is an ordering by definition 1.