Possible Duplicates:
Symmetric, Transitive and reflexive

Why isn't reflexivity redundant in the definition of equivalence relation?

Dependence of Axioms of Equivalence Relation?

Let $X$ a set and let $\sim$ a binary relation in $X$. $\sim$ is called a equivalence relation if:

  1. $\forall x\in X$ we have $x\sim x$.
  2. $\forall x,y\in X$ if $x\sim y$ then $y\sim x$.
  3. $\forall x,y,z\in X$ if $x\sim y$ and if $y\sim z$ then $x\sim z$.

I think that 1 is unnecessary because by 2 we have that $x\sim y \Leftrightarrow y\sim x$. Then by 3. we have that $x\sim y$ and $y\sim x$ then $x\sim x$. Then 2,3 $\Rightarrow$ 1.

Am I right?

  • $\begingroup$ Must I delete question or not? $\endgroup$ – Gaston Burrull Jun 1 '12 at 4:09
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    $\begingroup$ No; the question is simply closed, and a pointer to the duplicate is added. You should go read the answers there to see why you are not right. $\endgroup$ – Arturo Magidin Jun 1 '12 at 4:13
  • $\begingroup$ Yes I was full understood, then I found another duplicate "Dependence of Axioms of Equivalence Relation?" $\endgroup$ – Gaston Burrull Jun 1 '12 at 4:17
  • $\begingroup$ @ArturoMagidin In general titles are not explicit enough. Because this fact I didn't see a suggestive title. I always try to be very explicit in title as much as possible when i make a question. $\endgroup$ – Gaston Burrull Jun 1 '12 at 4:21
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    $\begingroup$ I found the first one by doing a search for reflexive transitive relation. $\endgroup$ – MJD Jun 1 '12 at 4:24

You are right unless there is some $x$ that is unrelated to the other elements. If $x\sim y$ is false for all $y$, then 2 and 3 might both hold, but 1 does not.

In particular, the empty relation, which has $x\not\sim y$ for all $x$ and $y$, is symmetric and transitive, but not reflexive.

  • $\begingroup$ What do you mean with empty relation? $\endgroup$ – Gaston Burrull Jun 1 '12 at 4:01
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    $\begingroup$ @Gastón: this is the relation in which nothing is related to anything else. (In other words, thinking of a relation on a set $X$ as a subset of $X \times X$, this is the empty subset of $X \times X$.) $\endgroup$ – Qiaochu Yuan Jun 1 '12 at 4:01
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    $\begingroup$ An empty relation is one in which $x\sim y$ is false for all $x$ and $y$. $\endgroup$ – MJD Jun 1 '12 at 4:02
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    $\begingroup$ Not only when the relation is empty: it is enough that there's some $\,x\in X\,$ that isn't related to any other element. Then we must require reflexivity $\endgroup$ – DonAntonio Jun 1 '12 at 4:03
  • $\begingroup$ @QiaochuYuan thanks then property 2 can't be used and can't deduce 1. $\endgroup$ – Gaston Burrull Jun 1 '12 at 4:03


what if there is no such $y$?

  • $\begingroup$ Do you mean for example the singleton? $\endgroup$ – Gaston Burrull Jun 1 '12 at 4:07

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