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Basically, we have 4 types of relations: reflexive, symmetric, antisymmetric, transtive. And then we separate 4 above types into 2 new definition:

  1. One relation is reflexive, symmetric, transitve called equivalent relation.

  2. One relation is reflexive, antisymmetrc, transitive called order relation.

All of them above are basic knowledge of elementary set theory.

So the question I wonder is "Does there exist one relation is both reflexive, symmetric, antisymmetric and transitive? If yes, so what is it called?"

Honestly, I have been finding out in the internet about my wonder, but of course I cannot see anything. Therefore, I post my question on here to ask everyone my question. Thanks for your helping.

Shaun
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VAKK_19
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  • What do you mean by anti-symmetric? Wouldn't it require a sign, an ordering? – Marius S.L. Oct 23 '21 at 18:48
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    Well, antisymmetric means $a\sim b$ and $b \sim a$ implies that $a=b$, and symmetric means that $a\sim b$ implies $b\sim a$, so..... – lulu Oct 23 '21 at 18:49
  • If it is both, $a \sim b \Longrightarrow b\sim a \Longrightarrow a=b$ then all equivalence classes (it is also transitive) have one element. And every singleton defines such a relation. So the answer is: yes, there is exactly one such relation, but it is a useless one. – Marius S.L. Oct 23 '21 at 19:10

1 Answers1

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Suppose $\sim$ is symmetric. Then for all elements $a,b$ in the ambient set $S$, we have that $a\sim b$ implies $b\sim a$. But if $\sim$ is antisymmetric, then $a\sim b$ and $b\sim a$ together imply $a=b$. Hence $\sim$ is in fact equality.

Shaun
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  • So if one relation is both symmetric and anti-symmetric, it is called "equality" relation? – VAKK_19 Oct 23 '21 at 19:05
  • Yes, @VAKK_19. What I mean is that, then, $\sim$ behaves precisely like $=$. – Shaun Oct 23 '21 at 19:06
  • wow I get it. But why does not any textbook I have found in the internet mention "equality" relation? That's so confusing. Or at least, why do not we claim "equality" relation as an official relation? If that we will have 5 types of relation: reflexive, symmetric, anti-symmetric, transitive and "equality". – VAKK_19 Oct 23 '21 at 19:11
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    @VAKK_19 Not only those two properties (symmetric and anti-symmetric). But the four properties you named in your question implies that $\sim$ is equality.. – jjagmath Oct 23 '21 at 19:12
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    "equality" is not a property of the relations. Equality IS a relation. – jjagmath Oct 23 '21 at 19:13
  • So in my exercise, I have proved one relation that satisfies all of 4 properties: reflexive, symmetric, anti-symmetric, transitive. What should I call that relation? Just write it is called "equality" relation? – VAKK_19 Oct 23 '21 at 19:16
  • Yes, @VAKK_19. Do you understand why? – Shaun Oct 23 '21 at 19:26
  • yes I understand why. But I am just confused about its name. I'm afraid my teacher will not accept that name because it is not in his textbook and his lecture...... – VAKK_19 Oct 23 '21 at 19:29
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    okay, I will believe in myself. Thank you for your help. – VAKK_19 Oct 23 '21 at 19:31
  • Sometimes it's known as the "diagonal relation", @VAKK_19 – Shaun Oct 23 '21 at 19:32
  • You're welcome, @VAKK_19. Please don't forget to accept this answer if it is satisfactory for you, by clicking the check mark. – Shaun Oct 23 '21 at 19:33