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

An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity.

Doesn't symmetry and transitivity implies reflexivity? Consider the following argument.

For any $a$ and $b$, $a R b$ implies $b R a$ by symmetry. Using transitivity, we have $a R a$.

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This is a good problem to pose in an introductory Discrete Math/Logic course. – Srivatsan Jul 31 '11 at 16:23

Actually, without the reflexivity condition, the empty relation would count as an equivalence relation, which is non-ideal.

Your argument used the hypothesis that for each $a$, there exists $b$ such that $aRb$ holds. If this is true, then symmetry and transitivity imply reflexivity, but this is not true in general.

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No.

The missing condition is sometimes called 'seriality' -- for any x there must be a y such that x R y.

If you add seriality to the symmetry and transitivity you get a reflexive relation again.

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I googled seriality, not much came up. You sure it's spelled right? – Chao Xu Jul 22 '10 at 10:40
Perhaps I made up the noun form. If you google 'serial binary relation' it will come up. Although if you google 'seriality of binary relations' it will come up there too. – bryn Jul 22 '10 at 10:42
@ChaoXu ProofWiki uses the name serial relation. – Martin Sleziak Jul 10 '12 at 12:37

It can be, if you use Euclideanity instead of transitivity: a R c and b R c implies a R b.

As an aside, Euclideanity plus seriality characterises the equivalence relations.

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 This seems to fail though because (as Akhil mentioned) the empty relation would satisfy this but it's not reflexive. – bryn Jul 22 '10 at 10:15 @bryn: ouch: somehow my eyes saw "reflexivity" and my brain read "symmetry". Different concepts, I have heard... – Charles Stewart Jul 22 '10 at 11:26 Not to sound as though I'm picking on you, but I think this revised claim is also false. My first clue was the fact that the wikipedia article you referenced said that Euclidean+Reflexive => Equivalence. Since being reflexive is a stronger requirement than being serial I sought a counterexample, the smallest of which I can find is on the set {1,2} and is defined as Rxy <-> y = 2. – bryn Jul 22 '10 at 11:57 @bryn: Rx2 & Rx2 -> Rxx. – Charles Stewart Jul 22 '10 at 12:14 @Charles: ouch: I typed it into a program that does a brute-force search for counterexamples, but I misrepresented the Euclideanity as 'Rxy & Rxz -> Ryz'. Perhaps I should have read the definition you gave! Sorry. – bryn Jul 22 '10 at 12:25
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