The notation of my teacher confuses me in the title. If x
is the same on the both sides, it seems trivial. So I suspect the x
is different so it becomes:
$x_{1}\leq x_{2}$ for every $x_{1} \in S$
so what it would mean is that every $x_{1} \in S$ has a maximum element $x_{2}$ that does not need be in $S$. Attacking the problem with an example, why not trivial with the explicit interpretation in the following. I can imagine a partial ordering that would satisfy the condition like cardinality with a $n$-cube where $n$ is the number of nodes and the number of nodes is $2^{n}$. To prove the condition in the case would require to show that the subsets form a chain with each nodes like the case with 3-cube $card(\emptyset) \leq card(\{1\}) \leq card(\{2\}) ... \leq card(\{1,2\}) \leq card(\{1,2,3\})$.
So what does the reflexive -term mean? Am I interpreting it right about maximum element or am I leaping to conclusion?