I am reading an article by Erick K. van Douwen (The Integers and Topology) in which the author mentions the decomposition of a set without definition. I've found a definition that describes it as a union of a disjoint family of sets, i.e., the decomposition of a set $X$ is a family:
$$\bigcup_{i \in I} A_{i}$$
of disjoint sets.
To me, there are two problems with this definition. Firstly, this seems to be the same definition of a partition of a set, I am confused if a partition and a decomposition of a set are equivalent. Secondly, in the context of the article, the following problem arises: Consider a set $[X]^{\omega}$ as the collection of subsets of $X$ with cardinality equals $\omega$. Now define a mad family on $X$ as the maximal (with respect to inclusion) quasi-disjoint subfamily of $[X]^{\omega}$.
We define a cardinal $\mathfrak{a}$ as follow:
$$\mathfrak{a} = \min\{|\mathcal{A}|:\text{there is a decomposition }\mathscr{D}\text{ of }\omega\text{ such that }\mathcal{A} \cap \mathscr{D} = \emptyset\text{ and such that }\mathscr{D} \cup \mathcal{A}\text{ is a mad on }\omega\}.$$
But if $\mathscr{D}$ is a subset of $\mathscr{P}(\omega)$ as well as $\mathcal{A}$, how can the conjuntion of this sets be empty?