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

Question

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?

Answer 1

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.