Is an reflexive relation also antisymetric?

discrete-mathematics relations

I've already seen a similar question here:

Is an Anti-Symmetric Relation also Reflexive?

But my question is rather, if you know that a relation is reflexive, then, can this relation also be antisymetric?

As far as I know, by definition, a relation is antisymetric if for two elements in R, xRy and yRx then x=y. For this to be true, I should only need x=y to be true for the relation to be antisymetric. In that case, the relation is antisymetric if we know it is reflexive already.

Am I wrong?


Solution 1:

Reflexivity does not imply anti-symmetry.

Consider a relationship defined over just $2$ elements: $a$ and $b$. And suppose that we have $aRa$, $aRb$, $bRb$, and $bRa$. Then $R$ is reflexive (since we have both $aRa$ and $bRb$), but $R$ is not anti-symmetric: we have $aRb$ and $bRa$, but it is not the case that $a = b$

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