Let $A$ be a set, $R$ an empty relation on $A$, what is $A/R$?

Question

Let $A=\{0,1,2\}$ be a set and $R=\{\}$. I know that $R$ is not an equivalence relation, but does it have to be? What is $A/R$ if $R$ is empty?

Examples:

$R_1=\{(0,0),(1,1),(2,2)\}$, $A/R_1=\{[0], [1], [2]\}$

$R_2=\{(0,0),(1,1),(0,1),(2,2)\}$, $A/R_2\{[0, 1], [2]\}$

$R=\{\}$, $A/R=?$

Answer 1

$R$ is not an equivalence relation, so $A/R$ does not make sense with the standard definition.

The extension suggested by user1 would probably be the most sensible in general: we can define $A/R$ to be $A/S$ where $S$ is the smallest equivalence relation containing $R$. In case of $R=\emptyset$ it would be the equality relation, so $A/R=\{\{a\}\mid a\in A\}$.

Answer 2

We know that the classes of $A/R$ form a partition of the set $A$ i.e. a set of disjoint non empty sets which their union is $A$ but if $R$ is the empty set speak of a partition does not make sense so $A/R$ does not exist.