Symmetric Preordering is Equivalence Relation

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: $5$
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