Trivial Ordering is Universally Compatible

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Theorem

Let $S$ be a set.

Let $\mathcal R$ be the trivial ordering on $S$.


Then $S$ is universally compatible.


Proof

To prove that the trivial ordering is in fact an ordering, we need to checking each of the criteria for an ordering:


Reflexivity

$\forall a \in S: a \mathcal R a$:


From its definition, we have $\forall a, b \in S: a = b \implies a \mathcal R b$.

Thus, as $a = a$, we have $\forall a \in S: a \mathcal R a$.

So reflexivity is proved.


Transitivity

$\forall a, b, c \in S: a \mathcal R b \land b \mathcal R c \implies a \mathcal R c$:


From the definition:

  • $a \mathcal R b \iff a = b$
  • $b \mathcal R c \iff b = c$

So as $a = b \land b = c \implies a = c$ from transitivity of equals, we have that $a \mathcal R c$ and thus transitivity is proved.


Antisymmetry

$\forall a, b \in S: a \mathcal R b \land b \mathcal R a \implies a = b$:


From the definition:

  • $a \mathcal R b \iff a = b$.
  • $b \mathcal R a \iff b = a$.

Antisymmetry follows from symmetry of equals.


The trivial ordering is by definition the same as the diagonal relation, and is therefore universally compatible.

$\blacksquare$