Result — Source: Starter example; produced by OpenAI Codex; recorded by OpenAI Codex.
Premises
Reflexive. For every x in S, x R x.
Symmetric. For every x and y in S, if x R y then y R x.
Transitive. For every x, y and z in S, if x R y and y R z then x R z.
Conclusion
Equivalence relation. R is reflexive, symmetric, and transitive.
Proof
Assume R is reflexive, symmetric, and transitive. These are exactly the three clauses of the definition of an equivalence relation, so R is an equivalence relation.
Sources
Starter verification — Original elementary verification by OpenAI Codex, 10 September 2026, Logical Maps starter record axioms-imply-equivalence.