Symmetric Preordering is Equivalence Relation
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Theorem
Let $\RR \subseteq S \times S$ be a preordering on a set $S$.
Let $\RR$ also be symmetric.
Then $\RR$ is an equivalence relation on $S$.
Proof
By definition, a preordering on $S$ is a relation on $S$ which is:
- $(1): \quad$ reflexive
and:
- $(2): \quad$ transitive.
Thus $\RR$ is a relation on $S$ which is reflexive, transitive and symmetric.
Thus by definition $\RR$ is an equivalence relation on $S$.
$\blacksquare$
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 7$: Exercise $7$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: Further exercises: $5$