to find the smallest and largest number of equivalence relation in a set

Question

Let $s$ be a set of $n$ elements. The number of ordered pairs in the largest and smallest equivalence relation on set $s$ are $n^2$ and $n$.

I am able to understand the largest set of equivalence relation, but in case of smallest set of equivalence relation it could be an empty set..so according to me it is 0.

Am I missing something?

Answer 1

The smallest equivalence relation must always contain the diagonal $\{(x,x) : x \in s \}$, because every element must be equivalent to itself. And the diagonal is an equivalence relation (equality) and has size $n$.

Answer 2

$S$ is a set of $n$ elements. $S$ is non-empty and we know, every non-empty set always contains the identity relation. So , It can't be empty relation , I suppose since empty relation is invalidated , that explains why smallest equivalence relation on set $S$ is $n$.

i.e Empty relation is transitive and symmetric on every set but not equivalence.