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.
Best Answer
For any set $A$, there exists only one relation which is both reflexive, symmetric and assymetric, and that is the relation $R=\{(a,a)| a\in A\}$.
You can easily see that any reflexive relation must include all elements of $R$, and that any relation that is symmetric and antisymmetric cannot include any pair $(a,b)$ where $a\neq b$. So already, $R$ is your only candidate for a reflexive, symmetric, transitive and antisymmetric relation.
Since $R$ is also transitive, we conclude that $R$ is the only reflexive, symmetric, transitive and antisymmetric relation.