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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?

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marked as duplicate by MJD, Qiaochu Yuan Jun 1 '12 at 4:06

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Must I delete question or not? –  Gastón Burrull Jun 1 '12 at 4:09
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. –  Arturo Magidin Jun 1 '12 at 4:13
Yes I was full understood, then I found another duplicate "Dependence of Axioms of Equivalence Relation?" –  Gastón Burrull Jun 1 '12 at 4:17
@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. –  Gastón Burrull Jun 1 '12 at 4:21
I found the first one by doing a search for reflexive transitive relation. –  MJD Jun 1 '12 at 4:24

2 Answers 2

up vote 5 down vote accepted

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.

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What do you mean with empty relation? –  Gastón Burrull Jun 1 '12 at 4:01
@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 $X \times X$.) –  Qiaochu Yuan Jun 1 '12 at 4:01
An empty relation is one in which $x\sim y$ is false for all $x$ and $y$. –  MJD Jun 1 '12 at 4:02
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 –  DonAntonio Jun 1 '12 at 4:03
@QiaochuYuan thanks then property 2 can't be used and can't deduce 1. –  Gastón Burrull Jun 1 '12 at 4:03


what if there is no such $y$?

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Do you mean for example the singleton? –  Gastón Burrull Jun 1 '12 at 4:07

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