# Am I correct? State the necessary and sufficient condition for R to be an equivalence relation on A.

Let R be a relation on a given nonempty set A. State the necessary and sufficient condition for R to be an equivalence relation on A.

My attempt

The conditions for any equivalence relation are Transitivity, Symmetric and Reflexive.

Reflexive : For all elements $x\in A$, $xRx$. Therefore, this is a necessary condition.

Symmetric : If $aRb$, then $bRa$. Since this is conditional, it should be a sufficient condition.

Transitivity : If $aRb$ and $bRc$, then $aRc$. Since this is conditional, it should be a sufficient condition.

Am I correct?

• This is the definition of an equivalence relation, rather than a "necessary and sufficient" condition. Is there more context to the exercise ? – beauby Nov 15 '12 at 6:07
• Suppose transitivity was a sufficient condition. Then if a relation is transitive, it is an equivalence relation. The relation "is taller than" is transitive: if Jim is taller than Bob and Bob is taller than Frank then Jim is taller than Frank. It is not, however, reflexive, which you've claimed is a necessary condition. Contradiction. We have a problem. – crf Nov 15 '12 at 6:08
• Incidentally, I'm not sure that your distinction between conditional and unconditional statements makes very much sense. – crf Nov 15 '12 at 6:09
• I copied the question as it exactly was. it was a finals question, and no extra context. – Yellow Skies Nov 15 '12 at 6:09
• @SingaporeanDude. Well, for $R$ to be reflexive, symmetric and transitive is, strictly speaking, necessary and sufficient for it to be an equivalence relation, but I find the wording a bit weird. – beauby Nov 15 '12 at 6:10