example of preorder on a set

Question

A binary relation $\preceq$ on $X$ is a preorderr if

1) it is reflexive, i.e. $x \preceq x$ for $x \in X$;

2) it is transitive, i.e. $x_{1}, x_{2}, x_{3} \in X$, $x_{1} \preceq x_{2}$ and $x_{2} \preceq x_{3}$ imply $x_{1} \preceq x_{3}$;

Unlike the simple order, the preorder (quasiorder) is not necessary antisymmetric (antisymmetric means $x_{1}, x_{2} \in X$, $x_{1} \preceq x_{2}$ and $x_{2} \preceq x_{1}$ imply $x_{1} = x_{2}$) and two elements can be noncomparable.

Could you, please, give an example of a preorder?

Answer 1

Every partial order is by definition a quasi-order as well, since a quasi-order is not necessarily anti-symmetric, but nowhere we require that it is not anti-symmetric either.

A more useful example would be the following relation on subsets of $\Bbb N$: $A\subseteq^* B$ if and only if $A\setminus B$ is finite. Namely, $A$ is "almost a subset" of $B$ if up to a finite mistake, it is indeed a subset. I will let you verify that it is indeed a quasi-order, and of course it is not anti-symmetric, since for every finite set $A$, $A\subseteq^*\varnothing\subseteq^* A$.

Answer 2

The integers under divisibility. More generally, for any integral domain, the divisibility relation is a quasi order which is a antisymmetric iff the unit group is trivial.