I am reading that $(\mathbb{Z}, \leq )$ is a total ordered set. I understand how it satisfies reflexivity, antisymmetry, transitivity.
But it says that because for any $a,b \in \mathbb{Z}$, either $a \leq b$ or $b \leq a$.
So if $a \sim b \iff a \leq b$, doesn't that mean if I choose $a = 2$ and $b = 1$, then it is not true that every $a,b,$ is either $a \leq b$?
EDIT: I thought that when we define relation on $(\mathbb{Z}, \leq )$, we define it as $$a \sim b \iff a \leq b$$ and not $$a \sim b \iff a \leq b \;\text{ or } \;b \leq a$$
If you pick $a=2$ and $b=1$, then it may not be true that $a\le b$, but it is that $b\le a$, and for it to be total ordered it is required that either $a\le b$ or $b\leq a$, not both (except in $a=b$ situation).
As far as relation being $a\leq b$ and not $b\leq a $, it is important to note that $a$ and $b$ are just placeholders and can be interchanged. The relation itself can be thought of as just $\leq$.
Edit: noticed after posting that this was already stated in the comments.