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

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  • $\begingroup$ 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). $\endgroup$ Mar 15, 2020 at 0:28
  • $\begingroup$ @EeveeTrainer This is from MIT OCW Analysis 1 Problem set 1. Link: ocw.mit.edu/courses/mathematics/18-100b-analysis-i-fall-2010/… $\endgroup$
    – DevrimA
    Mar 15, 2020 at 0:33
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    $\begingroup$ Please, write down the assignment and the two definitions, instead of using pictures. $\endgroup$
    – user239203
    Mar 15, 2020 at 0:43

2 Answers 2

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No, consider the relation $x\preceq y\stackrel{\text{def}}\iff x=y$ on a set with at least two elements.

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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.

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