Let $A=\{1,2,\ldots,n\},n\ge1$. Find the number of all relations $R\subseteq A\times A$ which are both both symmetric and antisymmetric.
Okay, so a relation is symmetric if $(\forall a\forall b)(aRb\Rightarrow bRa)$ and a relation is antisymmetric if $(\forall a\forall b)(aRb \land bRa\Rightarrow a=b).$
Graphically, a relation is symmetric if from the existence of the black arrow we can conclude that the red arrow (relation) also exists (and vice versa). But exactly this diagram is not allowed when we have an antisymmetric relation.
So I am a little confused and don't know how to count them. Thanks!