A basic question on the definition of order In the first chapter of Rudin's analysis book "order" on a set is defined as follows :
Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following two properties :
(i) If $x \in S$ and $y \in S$ then one and only one of the statements 
          $$ x < y, x=y, y<x $$ is true.
(ii) If $x,y,z \in S$, then $x < y$ and $y < z$ implies $x<z$. 
How is this different from the usual partial/total order notation. This looks like total order. Why is defining "order" like this ? Moreover, he has not defined $=$ here.

For the =, you can define it by negating the other two cases.  That is, = means not $<$ and not $>$
