\begin{definition}[Definition:Partitioning]
Let $S$ be a set.
Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$ such that:
:$(1): \quad \forall i \in I: S_i \ne \O$, that is, none of $S_i$ is empty
:$(2): \quad \ds S = \bigcup_{i \mathop \in I} S_i$, that is, $S$ is the union of $\family {S_i}_{i \mathop \in I}$
:$(3): \quad \forall i, j \in I: i \ne j \implies S_i \cap S_j = \O$, that is, the elements of $\family {S_i}_{i \mathop \in I}$ are pairwise disjoint.
Then $\family {S_i}_{i \mathop \in I}$ is a '''partitioning''' of $S$.
The image of this '''partitioning''' is the set $\set {S_i: i \in I}$ and is called a partition of $S$.
Note the difference between:
:the '''partitioning''', which is an indexing function (that is a mapping)
and
:the partition, which is the effect (that is, the image) of that mapping.
\end{definition}