Definition:Partitioning

Definition
Let $S$ be a set.

Let $\left \langle{S_i}\right \rangle_{i \in I}$ be a family of subsets of $S$ such that:
 * $(1): \quad \forall i \in I: S_i \ne \varnothing$, that is, none of $S_i$ is empty
 * $(2): \quad \displaystyle S = \bigcup_{i \mathop \in I} S_i$, that is, $S_i$ is the union of $\left \langle{S_i}\right \rangle_{i \in I}$
 * $(3): \quad \forall i, j \in I: i \ne j \implies S_i \ne S_j$, that is, the elements of $\left \langle{S_i}\right \rangle_{i \in I}$ are pairwise disjoint.

The image of the partitioning is the set $\left\{{S_i: i \in I}\right\}$ and is called a partition of $S$.

Note the difference between:
 * the partitioning, which is a family (that is a mapping)

and
 * the partition, which is the effect (that is, the image) of that mapping.