Reflexive Relation on Singleton is Well-Ordering
Jump to navigation
Jump to search
Theorem
Let $S = \set s$ be a singleton.
Let $\RR$ be a reflexive relation on $S$.
Then $\RR$ is a well-ordering on $S$.
Proof
Let $S = \set s$.
By definition of reflexive relation:
- $s \mathrel \RR s$
It trivially holds that:
- $\forall a, b \in S: a \mathrel \RR b \land b \mathrel \RR a \implies a = b$
and so $\RR$ is antisymmetric.
It also trivially holds that:
- $\forall a, b, c \in S: a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c$
Thus $\RR$ is an ordering on $S$.
We also have trivially that:
- $\forall a, b \in S: a \mathrel \RR b \lor b \mathrel \RR a$
and so $\RR$ is a total ordering on $S$.
Finally from Finite Totally Ordered Set is Well-Ordered:
- $\struct {S, \RR}$ is a well-ordered set.
Hence the result.
$\blacksquare$