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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.

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  • $\begingroup$ What do you mean by anti-symmetric? Wouldn't it require a sign, an ordering? $\endgroup$ Oct 23, 2021 at 18:48
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    $\begingroup$ 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..... $\endgroup$
    – lulu
    Oct 23, 2021 at 18:49
  • $\begingroup$ 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. $\endgroup$ Oct 23, 2021 at 19:10

1 Answer 1

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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.

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  • $\begingroup$ So if one relation is both symmetric and anti-symmetric, it is called "equality" relation? $\endgroup$
    – VAKK_19
    Oct 23, 2021 at 19:05
  • $\begingroup$ Yes, @VAKK_19. What I mean is that, then, $\sim$ behaves precisely like $=$. $\endgroup$
    – Shaun
    Oct 23, 2021 at 19:06
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    $\begingroup$ @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.. $\endgroup$
    – jjagmath
    Oct 23, 2021 at 19:12
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    $\begingroup$ "equality" is not a property of the relations. Equality IS a relation. $\endgroup$
    – jjagmath
    Oct 23, 2021 at 19:13
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    $\begingroup$ okay, I will believe in myself. Thank you for your help. $\endgroup$
    – VAKK_19
    Oct 23, 2021 at 19:31

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