Can there be a relation which is reflexive, symmetric, transitive, and antisymmetric at the same time? I tried to find so.
If $A = \{ a,b,c \}$. Let $R$ be a relation which is reflexive, symmetric, transitive, and antisymmetric.
$R = \{ (a,a), (b,b), (c,c) \}$
Is this correct? If I'm wrong, can you help me understand it?
Since if $(a, b)$ and $(b, c)$ are elements of $R$ by transitive there would be $(a, c)$, but then there should be $(b, a)$, $(c, b)$ and $(c, a)$ by symmetry, but then it would not be antisymmetric. If I'm not mistaken.