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1.1 Definition. A set $D$ is a directed set if there is an order relation $\prec$ defined on $D$ which satisfies the following:

1.$\prec$ is reflexive

2.$\prec$ is transitive

3.For $x,y\in D,\exists z \in D$ s.t. both $x \prec z$ and $y \prec z$

Which of the following sets are directed.?

  1. $\Bbb R$ with $<$

  2. $\Bbb R$ with $\leq$

  3. $P(X)$ for $X \ne \emptyset$ with $\prec$ means “subset of” instead of precedes

I am guessing (2) and (3) are directed

I am new to dealing with this topic, so am unsure of what to do .

jjagmath
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Dazed and Confused
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  • Your definition of directed set has a little mistake. The relation is not necessarily an "order" since it antisymmetry is not required. – jjagmath Sep 16 '22 at 02:03
  • " so am unsure of what to do ." If you are guessing that (1) is not directed you give a reason for it: which of the properties $<$ fails to satisfy. If you are guessing that (2) and (3) are directed sets, you need to prove that the 3 conditions on the respective relations (and sets) are satisfied. – jjagmath Sep 16 '22 at 02:09
  • So I have to prove it. The exercise doesn’t ask for proof – Dazed and Confused Sep 16 '22 at 14:14
  • In Mathematics we prove things. So unless this is an exercise from another field, yes, you're expected to prove what you claim. – jjagmath Sep 16 '22 at 21:37
  • I seen examples of this,but how to prove a directed none. I was hoping I can get an example proof of one of them so I can carry on – Dazed and Confused Sep 17 '22 at 00:55

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