In Rudin, Principles of Mathematical Analysis (ed. 3), he provides the following definition (pp. 3)
Definition: Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following two properties:
(1) If $x \in S$ and $y\in S$ then one and only one of the statements: $x<y$, $x=y$, or $y<x$ is true.
(2) If $x,y,z \in S$, if $x<y$ and $y<z$, then $x<z$
The question I asked myself is if property 2 must hold by definition. There answer I had come up with is yes.
Suppose that the above definition is provided with only the first condition. Now take a set, $S=(1,3,5)$. Next define the following order on $S$: $1<3$, $3<5$, $5<1$.
The above order satisfies the first property of the defintion, but not the second. Hence Transitivity as a definition is required.
Is my reasoning correct, and is there anything I missed?