# Establishing whether or not a set is ordered

I'm having trouble with determining whether the first criterion is met or not, specifically in the case of $$x. Could someone please help me out here?

• I'm confused, is there like an example/homework problem/exercise you're working on or something? If so, the example set you're working with is important to include (it'll make life easier for us if nothing else). Mar 15, 2020 at 0:28
• @EeveeTrainer This is from MIT OCW Analysis 1 Problem set 1. Link: ocw.mit.edu/courses/mathematics/18-100b-analysis-i-fall-2010/… Mar 15, 2020 at 0:33
• Please, write down the assignment and the two definitions, instead of using pictures.
– user239203
Mar 15, 2020 at 0:43

No, consider the relation $$x\preceq y\stackrel{\text{def}}\iff x=y$$ on a set with at least two elements.
Let $$X$$ be the set of all subsets of $$\Bbb N$$ and define $$x \preceq y$$ iff $$x \subseteq y$$, a classical example of a partially ordered set. If we define $$\prec$$ by the recipe indicated it will not be an order (as defined in 1.5) because e.g. $$\{1\}$$ and $$\{2\}$$ have no relationship, they're different and neither is a subset of the other. Many such pairs exist, of course.