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I've been wondering recently the following:

Let $\sim $ be a symmetric and transitive relation defined on $S $. Let $a \sim b $, which implies $b \sim a $ by symmetry, and by transitivity, $a \sim a $. Hence, $\sim $ is reflexive.

I can't think of any counterexamples, although any are welcome. Also, if there is some counterexample, where could be the flaw in my reasoning be?

Andrew Tawfeek
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2 Answers2

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You're assuming that there exists a $b$ such that $a\sim b$ in your argument. If you don't know the relation is reflexive, that $b$ may not exist. Put differently, reflexivity in the presence of symmetry and transitivity is equivalent to each element being equivalent to some other one

Louis
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  • Ohhhh I think I understand, if $\sim $ is not reflexive, then the equivalence class $[a]_\sim $ may as well be empty because there's no guaranteed element in that set? – Andrew Tawfeek Jan 21 '17 at 00:24
  • Hmmm.. So is that error what makes the empty relation a counterexample? – Andrew Tawfeek Jan 21 '17 at 00:25
  • The empty relation is a little confusion to think about. Here's a clearer counterexample: just take any set and define an equivalence relation on a strict subset. E.g. on the set of all people, let two people (the set) be equivalent if they have the same age and also are both blond (the strict subset). Then that's symmetric and transitive, but not reflexive because brown haired people are not in relation to anything.. – Louis Jan 23 '17 at 17:20
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Reflexivity is a~a. You are using symmetry as reflexivity. As vadim123, noted, symmetry and transitivity do not imply reflexivity. The empty relation is a counterexample.

  • Oh thank you very much. I did not think of that! I'll edit the answer –  Jan 20 '17 at 23:01
  • If you included a "for all $a\in S$" in the first sentence, it would make it easier to upvote. – Lukas Betz Jan 20 '17 at 23:01
  • I actually had just finished editting my post, accidentally mixed up the terms, but yes, that's what I meant. And is that above a sufficent "proof" then? – Andrew Tawfeek Jan 20 '17 at 23:03
  • @vadim123 doesn't empty relation satisfy reflexivity (and all other things) vacuously? – user160738 Jan 20 '17 at 23:03
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    You can read a complete argument in here: http://math.stackexchange.com/q/1081333/194469 –  Jan 20 '17 at 23:05
  • Could you (just roughly) explain what the empty relation is? And is that the only counterexample? – Andrew Tawfeek Jan 20 '17 at 23:06
  • Oh, okay, I was thinking of "relation on $\emptyset$". @AndrewTawfeek empty relation is just an empty set of ordered pairs – user160738 Jan 20 '17 at 23:07