# Antisymmetric relation (“strong” vs “weak”)

Defining:
"weak antisymmetric relation": $\forall a, b \left< a,b \right> \in R \land \left< b,a \right> \in R \Rightarrow a=b$

"Strong antisymmetric relation": $\forall a, b \left< a,b \right> \in R \Rightarrow \left< b,a \right> \notin R$

I've read that "strong" is in particular a "weak" relation, but as I understand:
"strong" isn't reflexive while "weak" is. So, how can "strong" be in particular "weak"?

I hope what I'm asking is understandable.
Thanks.

• Not every weak relation is reflexive. – mrkvon Oct 14 '19 at 10:49

It's useful to think of a weak antisymmetry as $\leq$, for whatever intuitive meaning of $\leq$ you're more comfortable with (for example the order relation on the real numbers). Similarly you can think of a strong antisymmetry as being $<$.