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I've come across a relation that satisfies all four of the relation properties: reflexive, symmetric, antisymmetric, and transitive.

First of all, is this even possible?? and if so, is it a very specific kind of relation?

The following is the relation:

Let A = {−7, −5, −3, −1, 1, 3, 5, 7}

R ={(a,b)|a^3 =b^3}

Any help is appreciated.

  • If a relation is symmetric and antisymmetric, then the set only has one element (prove this!). I don't think your relation works for all four. – Sean Roberson Apr 11 '18 at 04:07
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    @SeanRoberson The relation $=$ (which is what the OP is asking us about) has all four properties. – bof Apr 11 '18 at 04:11
  • Hint: $,a^3=b^3 \iff a=b,$ for real numbers. – dxiv Apr 11 '18 at 04:13
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    @SeanRoberson A relation is symmetric and antisymmetric if and only if it is a subset of the equality relation. – David Apr 11 '18 at 04:16
  • @David a strict subset of the equality relationship would violate the Reflexive property. – Q the Platypus Apr 11 '18 at 04:16
  • @QthePlatypus Yes I know, I have already modified my comment. – David Apr 11 '18 at 04:16
  • @David so is this an equivalence relation? –  Apr 11 '18 at 04:17
  • @ShaunMoini Yes, because (I assume) you have already proved it is reflexive, symmetric and transitive. – David Apr 11 '18 at 04:19
  • @David correct, but i have also proved that it is antisymmetric. –  Apr 11 '18 at 04:20
  • As to the question of if a relation can simultaneously be reflexive, symmetric, antisymmetric, and transitive, see this related question. – JMoravitz Apr 11 '18 at 04:20
  • @ShaunMoini True but irrelevant. An equivalence relation is reflexive, symmetric and transitive, and may or may not have other important properties besides. – David Apr 11 '18 at 04:21
  • @David then how would I be able to tell apart an equivalence relation vs. a partial order if all of the properties are satisfied? –  Apr 11 '18 at 04:23
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    @ShaunMoini Your example is both an equivalence relation and a partial order. It is (essentially) the only possible example of this. – David Apr 11 '18 at 04:24
  • @ShaunMoini However equality is a very unusual example of a partial order and IMHO it's better to think of it as a "fluke" rather than a "typical" partial order. – David Apr 11 '18 at 04:25
  • @David THANK YOU –  Apr 11 '18 at 04:26

1 Answers1

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The equality relationship (also called the identity relationship) has all four properties.

reflexive: $\forall x \in X : x = x$

symmetric: $\forall a,b \in X : a = b \iff b = a$

antisymmetric: $\forall a, b \in X: (a=b) \wedge (b=a) \implies a = b$

transitive: $\forall a, b, c \in X: (a=b) \wedge (b=c) \implies a = c$

Q the Platypus
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