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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?

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    Must I delete question or not?2012-06-01
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    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.2012-06-01
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    Yes I was full understood, then I found another duplicate "Dependence of Axioms of Equivalence Relation?"2012-06-01
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    @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.2012-06-01
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    I found the first one by doing a search for `reflexive transitive relation`.2012-06-01
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    The explicit title is the second and third. The first one is quite not suggestive.2012-06-01
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    The search also shows the first few lines of the body of the post. In the one I found, the first few words were "If a relation is symmetric and transitive, then it will be reflexive too. True/False?"2012-06-01

2 Answers 2

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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?2012-06-01
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    @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 $X \times X$.)2012-06-01
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    An empty relation is one in which $x\sim y$ is false for all $x$ and $y$.2012-06-01
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    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 reflexivity2012-06-01
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    @QiaochuYuan thanks then property 2 can't be used and can't deduce 1.2012-06-01
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    @DonAntonio Yes you're right if there is some $x$ not related to anyone then can't be deduce that $x\sim x$.2012-06-01
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    Is surprising that the answer in this question was opposite to my other question about definition of topology http://math.stackexchange.com/questions/151924/unnecessary-property-in-definition-of-topological-space2012-06-01
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no

what if there is no such $y$?

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    Do you mean for example the singleton?2012-06-01