Set families/Introduction/Section

Not only elements but also sets can be indexed by an index set. This is called a family of sets.

Definition

Let ${}I$ be a set, and let, for every ${}i\in I$, a set ${}M_{i}$ be given. Such a situation is called a family of sets

$M_{i}{\text{ , }}i\in I.$

The set ${}I$ is called the index set of the family.

Here, the sets might be independent of each other, but they can also be subsets of a certain set.

Definition

Let ${}M_{i}$, ${}i\in I$, be a family of subsets of a set ${}G$. Then

${}\bigcap _{i\in I}M_{i}={\left\{x\in G\mid x\in M_{i}{\text{ for all }}i\in I\right\}}\,$

is called the intersection of the sets, and

${}\bigcup _{i\in I}M_{i}={\left\{x\in G\mid {\text{ there exists some }}i\in I{\text{ with }}x\in M_{i}\right\}}\,$

is called the union of the sets.

Definition

Let ${}I$ be a set, and let, for every ${}i\in I$, a set ${}M_{i}$ be given. Then the set

${}M=\prod _{i\in I}M_{i}=\{(x_{i})_{i\in I}:\,x_{i}\in M_{i}{\text{ for all }}i\in I\}\,$

is called the product set of the ${}M_{i}$.

As soon as one of the sets ${}M_{i}$ is empty, then also the product is empty, because then there is no possible value for the ${}i$-th component. However, if all sets ${}M_{i}$ are not empty, then also their product is not empty, as for every index ${}i$, there exists at least one element ${}x_{i}\in M_{i}$. In a formal-axiomatic introduction of set theory, one has to postulate that such a choice is possible. This is the content of the axiom of choice.

Example

For ${}n\in \mathbb {N}$, let

${}M_{n}={\left\{x\in \mathbb {N} \mid x\geq n\right\}}\,$

be the set of all natural numbers that are at least as large as ${}n$. This is a family of subsets of ${}\mathbb {N}$ indexed by the natural numbers. We have the inclusions

$M_{n}\subseteq M_{m}{\text{ for }}n\geq m.$

The intersection

$\bigcap _{n\in \mathbb {N} }M_{n}$

is empty because there is no natural number that is above every other natural number.

Example

For ${}n\in \mathbb {N} _{+}$, let

${}\mathbb {N} n={\left\{x\in \mathbb {N} _{+}\mid x{\text{ is a multiple of }}n\right\}}\,$

be the set of all positive natural numbers that are multiples of ${}n$. This is a family of subsets of ${}\mathbb {N}$ indexed by the positive natural numbers. We have the inclusions

$\mathbb {N} n\subseteq \mathbb {N} m{\text{ for }}m{\text{ divides }}n.$

The intersection

$\bigcap _{n\in \mathbb {N} _{+}}\mathbb {N} n$

is empty because no positive natural number is a multiple of every positive natural number (${}0$ is such a multiple).

Example

Let ${}x$ be a real number, and let ${}x_{n}$ denote the rational number that consists of the digits of ${}x$ in the decimal system up to the ${}n$th digit after the point. We define the intervals

${}M_{n}=\left[x_{n},x_{n}+{\left({\frac {1}{10}}\right)}^{n}\right]\subset \mathbb {R} \,.$

These are intervals of length ${}{\left({\frac {1}{10}}\right)}^{n}$, and we have

${}\{x\}=\bigcap _{n\in \mathbb {N} }M_{n}\,.$

The family ${}M_{n}$, ${}n\in \mathbb {N}$, is a family of nested intervals for ${}x$.