Is the relation on a singleton set an equivalence relation?

Question

So I understand that for a relation on a set to be an equivalence relation, it must satisfy three axioms:

For all $x, y, z \in X$ and the relation $R$ on $X$,

1. $(x,x) \in R$
2. if $(x,y) \in R$ then $(y,x) \in R$
3. if $(x,y) \in R$ and $(y,z) \in R$ then $(x,z) \in R$

So if you have a relation on a singleton for example $ \{0\} $ then that relation would be $\{(0,0)\}$

Therefore, the first axiom is obviously satisfied. However would the rules for symmetry and transitivity be satisfied as well? Or do you need $x \ne y \ne z $?

Answer 1

Yes, they are satisfied for the relation you specified. (As pointed out in comments there is another relation though, the empty relation, that is not reflexive.) There is no assumption made that $x,y,z$ are distinct. Indeed, even if there is more than one element the implications also need to hold in the case where some or all of $x,y,z$ are equal.

In another direction, if you have no instance that fulfills the premise the implication is still true.

So, if you had a condition of say "unequal symmetry" saying: "if for $x \neq y$ we have $(x,y) \in R$ then $(y,x) \in R$" then your relation would also have this property.