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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?

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    @crf Interesting breakdown of the question! I see so many errors in my logic~~2012-11-15

1 Answers 1

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A definition provides a condition which is both necessary and sufficient. So when we write,

Definition. An equivalence relation on a nonempty set is a relation which is reflexive, symmetric, and transitive.

We are saying that this is a single condition which is both necessary and sufficient for a relation to be considered an equivalence relation. We can break it up into three different conditions if we like, each of which would also be both necessary and sufficient.

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    Well, I mean, it's not false to say "the necessary and sufficient condition for a relation to be an equivalence relation is that it satisfies reflexivity, symmetry, and transitivity". But it is a strangely worded question—I would usually just ask for "the definition".2012-11-15