Definition:Trivial Partition

Definition
Let $S$ be a set such that $S \ne \varnothing$.

There are two partitions on $S$ which are referred to as the trivial partitions on $S$:

Singleton Partition
The singleton partition on $S$ is defined as:
 * $\mathcal P = \left\{{S}\right\}$

That is, it is a partition with only one component.

Partition of Singletons
The partition of singletons on $S$ is defined as:
 * $\mathcal P = \left\{{\left\{{x}\right\}: x \in S}\right\}$

That is, it is a partition such that every element of $\mathcal P$ is a singleton.