Given a partial order $\prec$ on a non-empty set $A$ and a set $B$, which is a non-empty subset of $A$, I have to prove, that $min B$ is unique (provided it exists).
My problem understanding this, is the following:
Let's say:
$A=\mathcal{P}(\{a,b\})=\{\emptyset,\{a\},\{b\},\{a,b\}\}$ and
$\{\{a\},\{b\},\{a,b\}\}=B\subseteq A$
$X\prec Y :\Leftrightarrow X\subseteq Y$
The requirements of the problem are satisfied, but $min B$ is not unique, or am I missing something?
I have no problem solving it for a total order, but I don't get how uniqueness is accomplished in a partial order.