Relation symmetric and antisymmetric

Question

Let $A$ a non-empty set. If there is a complete relation on $A$ that is both symmetric and antisymmetric, does it imply that the relation is the "equality" and $A$ has one single element?

Answer 1

$A$ can consist of any number of elements, but the relation must indeed be "equality" since the relation only satisfies reflexivity.

Antisymmetry: For all $x,y \in A$: $xRy$ and $yRx$ imply $x=y$

Symmetry: For all $x,y \in A$: $xRy$ implies $yRx$.

Taking both, we conclude $xRy$ implies $x = y$.