Sets and Indices

Definitions

A set whose elements are sets themselves is called a Family.
The elements of a family are called Members.
When each element of a set $\Gamma$ corresponds to a set $A_{\gamma}$ with $\gamma$ as the index, $\Gamma$ as the index set, and $\left\{ A_{\gamma} : \gamma \in \Gamma \right\}$ as the indexed family.

Explanation

Though the term ‘Family’ was originally introduced as a refined term for ‘set of sets’, this expression was merely to avoid the uncomfortable notion of a ‘set of sets’; yet, it still remains counterintuitive. Creating new terminology just to denote a set having sets as elements is odd. It implies that the term Family is solely introduced for convenience.

Consider the following family as an example: $$ \mathcal{F}=\left\{\left\{ 1 \right\} , \mathbb{R} , \mathbb{Q}, \emptyset , \mathbb{R} \right\} $$ $\mathcal{F}$ can be called a family since all its elements are sets. It is noteworthy that $\mathbb{R}$ is used repeatedly. This notation indicates that a family is not just a simple set of sets, nor is it strictly a set of sets in the precise sense. It is strictly for convenience. In that context, the term family cannot derive anything other than Family from the English expression, failing to capture its intended meaning. Even the term ‘member’ is refined as ‘component’, which is not a convenient expression either, hence this blog will retain the terms Family and Member.

Similarly, indexing as ‘subscript’ is overly refined. Let’s construct an indexed family for $\mathcal{F}$ as shown above. Fortunately, being a finite set, if we set $$ A_{1} = \left\{ 1 \right\} \\ A_{2} = \mathbb{R} \\ A_{3} = \mathbb{Q} \\ A_{4} = \emptyset \\ A_{5} = \mathbb{R} $$ for $\Gamma = \left\{ 1,2,3,4,5 \right\}$, we obtain $\mathcal{F} = \left\{ A_{\gamma} : \gamma \in \Gamma \right\}$. Note that while $A_{2} = \mathbb{R}$ and $A_{5} = \mathbb{R}$, this allowance for duplication is not to undermine the foundation of set theory but for the sake of convenient expression. For the same reason, a family is also referred to as a Collection. It might seem circular since in English, a set is defined using the term Collection, but as mentioned earlier, it’s all for the sake of convenience, so don’t overthink it; just follow the convention of the material you are working with.

On the other hand, indexes don’t necessarily have to maintain an order or be specifically numbered. Consider setting $\Gamma = \mathbb{R}$ and thinking of $A_{\gamma}$ as intervals $[k , k+1)$ over $k \in \mathbb{Z}$. Thus, $\gamma \in \mathbb{R}$ results in $A_{\pi} = [3,4)$, $A_{\sqrt{10}} = [3,4)$, making it possible to locate the corresponding $A_{\gamma}$ for every $\gamma \in \Gamma$. Such peculiar configurations might seem unnecessary, but take Topology, for example, where such sets are used as naturally as one breathes.

For any family $\mathcal{F}$, the following expressions are used:

Union: $$ \bigcup \mathcal{F} = \bigcup_{A \in \mathcal{F}} \left\{ x \in U : \exists A \in \mathcal{F} , x \in A \right\} $$
Intersection: $$ \bigcap \mathcal{F} = \bigcap_{A \in \mathcal{F}} \left\{ x \in U : \forall A \in \mathcal{F} , x \in A \right\} $$

Basic Properties

For $\left\{ A_{\gamma} : \gamma \in \Gamma \right\}$, the following holds:

[1] Form of the Principle of Inclusion: For the universal set $U$, $$ \bigcup_{\gamma \in \emptyset} A_{\gamma} = \emptyset \\ \bigcap_{\gamma \in \emptyset} A_{\gamma} = U $$
[2] Generalization of De Morgan’s Theorem: $$ \left( \bigcup_{\gamma \in \Gamma} A_{\gamma} \right)^{c} = \bigcap_{\gamma \in \Gamma} A_{\gamma}^{c} \\ \left( \bigcap_{\gamma \in \Gamma} A_{\gamma} \right)^{c} = \bigcup_{\gamma \in \Gamma} A_{\gamma}^{c} $$
[3] Distributive Law: About the set $B$, $$ \left( \bigcup_{ \gamma \in \Gamma } A_{\gamma} \right) \cap B = \bigcup_{\gamma \in \Gamma} \left( A_{\gamma} \cap B \right) \\ \left( \bigcap_{ \gamma \in \Gamma } A_{\gamma} \right) \cup B = \bigcap_{\gamma \in \Gamma} \left( A_{\gamma} \cup B \right) $$