Is there something wrong with the following two-axiom definition of an equivalence relation?
A relation $R \subseteq A\times A$ is an equivalence relation iff:
- For all $x \in A, xRx$
- For all $x, y, z \in A, xRy$ and $xRz$ implies $yRz$
Given $R \subseteq A \times A$ which satisfies (1) and (2), $R$ is reflexive by (1).
For $x, y \in A$ with $xRy$, we have $xRx$ so by (2) we have $yRx$ and $R$ is symmetric.
For $x, y, z \in A$ with $xRy$ and $yRz$ we have $yRx$ so by (2) we have $xRz$ and $R$ is transitive.
So $R$ is an equivalence relation.
Given an equivalence relation $R \subseteq A \times A$, (1) holds.
For any $x, y, z \in A$ with $xRy$ and $xRz$ we have $yRx$ so $yRz$ and (2) holds.
I have only seen somenthing similar to this in a wikipedia page about Tarski's axioms for euclidean geometry. If there is nothing wrong with the proof, why don't we prefer a two-axiom definition over a three-axiom one? The usual definition surely seems much more natural but I can't even find something on the internet for the second one. Is this something generally discused about equivalence relations (something like the dependance of the axioms of a metric space)?