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I was solving this question:

On a set S , $a\;R\;b$ if $a=b$ , prove it's an equivalence relation

Edited proof:

  1. $R = \{(x,x) \;|\; x \in S \} $

    since every element $x=x$ under the relation $R$ for all $x \in S$ , hence it's reflexive

  2. $ \forall\; (x,y) \in R$ , since $x=y$ then $y=x$ also, hence it's symmetric

  3. $ \forall\; (x,y),(y,z) \in R$ since $x=y$ and $y=z$ $=>$ $x=z$, hence it's transitive

I want to ask whether my proof is right? And if right did I do mathematically or more in layman's term?

Thanks

Edit: Now it seems R contains only one element $(a,a)$

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2 Answers

up vote 1 down vote accepted

1) What you wrote here is wrong: you must show

$$\forall\,x\in S\,\,,\,\,xRx\,\,\text{(or}\,\,(x,x)\in R)$$

and not $\,R=\{(x,x)\;:\;x\in S\}$

About the whole thing: you say $\,x=y\Longrightarrow y=x\,$ "by logic"...whose logic??

This, and the next transitivity point depend generally on what you can rely on: if you've studied and/or have been said the equality relation is symmetric/transitive then...well, there's nothing to prove in fact, is there? If you haven't then you'll have to use other tools that most probably were given to you in that course.

The point is: you didn't prove anything, you just wrote down the definitions of every characteristic of an equivalence relations.

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I must say I need hint now how to solve this , I'm totally baffled by question itself –  Mr.Anubis Sep 23 '12 at 14:16
    
I've edited my proof , can review it now? –  Mr.Anubis Sep 23 '12 at 14:23
    
@Mr.Anubis , I still can't see any difference with what you edited and the OP. Equality is either assumed to be an equivalence relation on any set or else some other porperties/conditions are imposed on some certain set. You msut realize that you haven't yet proved anything at all. –  DonAntonio Sep 23 '12 at 14:32
    
can you give me some hints then please? –  Mr.Anubis Sep 23 '12 at 15:40
    
I honestly can't @Mr.Anubis as if you're not given anything else I'd say the equality relation is an equivalence one per definition –  DonAntonio Sep 23 '12 at 15:56
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I totally agree with DonAntonio. You didn't prove anything here. In fact, information is insufficient to prove that $R$ is an equivalence relation. You need to specify some properties on set $S$.

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How do I prove then? There is no other data given , I myself was confused with the question only the relation = on set S is given –  Mr.Anubis Sep 23 '12 at 14:15
    
Please give a look at my edited proof –  Mr.Anubis Sep 23 '12 at 14:25
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