# Is an reflexive relation also antisymetric?

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?

• Reflexive is x=x not x=y. are you thinking of symmetric? Also your question seems confusing: Are you asking if a relation's being reflexive forces it to be antisymmetric? Dec 1 '21 at 1:12
• Yes that's the question, sorry if it's confusing. Dec 1 '21 at 1:18

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 $$a = b$$

• I see. I didn't consider a case like this. it seems that I was "mixing" the definitions of reflexive and anti-symetric. Silly mistake. Thanks for clarifying! @Bram28 Dec 1 '21 at 1:26
• @nutshell_A You're welcome! I make silly mistakes like this all the time :P Dec 1 '21 at 1:34